Phase-stabilized Ultrashort Laser Systems for Spectroscopy

Contents

Zusammenfassung

Abstract

1 Introduction
1.1 Time-resolved spectroscopy
1.2 Optical frequency metrology with frequency combs

2 Ultra-broadband oscillators
2.1 Few-cycle Kerr-lens mode-locked Ti:sapphire oscillator
2.2 Chirped mirror technology for dispersion control
2.3 The carrier-envelope phase of a mode-locked oscillator
2.3.1 Measurement of the frequency comb parameters
2.3.2 CE phase stabilization by difference frequency generation
2.3.3 Control of the frequency comb parameters
2.3.4 CE phase stability characterization
2.4 Long-cavity chirped-pulse oscillators
2.4.1 Double-pass post-amplifier
2.5 Conclusions

3 Few-cycle chirped-pulse amplifier systems
3.1 CE phase-stabilized chirped-pulse amplifier system
3.1.1 Origins of CE phase noise of amplified pulses
3.2 Conclusions

4 Femtosecond enhancement cavities
4.1 Passive optical resonators for femtosecond pulses
4.1.1 Dispersion control
4.1.2 Electronic feedback techniques
4.2 Vacuum enhancement cavity at 10 MHz repetition rate
4.3 Conclusions

5 Applications
5.1 Spectroscopy experiments with frequency combs
5.2 High-order harmonic generation
5.2.1 High harmonic generation from surfaces
5.2.2 High-order harmonic generation in an enhancement cavity ...
5.3 Above-threshold ionization
5.4 Conclusions

6 Outlook

A Appendix
A.1 Origin of the frequency comb
A.2 Offset frequency dependence on group and phase velocity

Bibliography

Acknowledgements

Curriculum Vitæ

Zusammenfassung

Die Erforschung der Wechselwirkung von Laserlicht und Materie erfordert laufend neue Technologien für immer kürzere Laserpulse, um atomare Prozesse in immer kleineren Dimensionen zeitaufgelöst und mit hoher Präzision verfolgen zu können. Solche Pulse bestehen meist nur noch aus einigen wenigen Zyklen des elektrischen Feldes. Daher spielt die Phase dieser Feldoszillationen mit kürzer werdender Pulsdauer eine immer größere Rolle. Ihre Stabilität liefert einen entscheidenden Beitrag für die Genauigkeit der Messung.

In dieser Arbeit werden mehrere Ultrakurzpuls-Laserquellen vorgestellt, die Pulse mit wenigen Zyklen und einem kontrollierten Verlauf des elektrischen Felds erzeugen. Zunächst wird eine Methode zur Phasenstabilisierung von Laseroszillatoren mit grosser Bandbreite diskutiert. Sie verbessert die Reproduzierbarkeit der Phase um eine Grössenordnung verglichen mit vorherigen Ansätzen.

Desweiteren wurde solch ein Oszillator in ein phasenstabilisiertes Verstärkersystem integriert. Die von diesem System erzeugten hochenergetischen phasenstabilisierten Laserpulse wurden in einer Reihe von Experimenten zur zeitaufgelösten Untersuchung von Starkfeldphänomenen eingesetzt. Beispielsweise wurde das laserinduzierte Tunneln von Elektronen aus einem Atom auf einer Zeitskala im SubFemtosekundenbereich beobachtet.

Eine weitere Messung, die nur durch einem kontrollierten Feldverlauf ermöglicht wurde, wird hier präsentiert: Einzelne Signaturen der elektrischen Feldhalbzyklen wurden im Spektrum von Photoelektronen identifiziert.

Die Frequenzkonversion von hochenergetischen Laserpulsen in hohe Harmonische ist ein gebräuchliches Verfahren, um kohärentes Licht im extremen ultravioletten (XUV) Spektralbereich zu erzeugen. Viele Anstrengungen wurden unternommen, um die niedrige Effizienz dieses nichtlinearen Prozesses zu steigern. Als ein potentiell möglicher Weg wird hier die Erzeugung hoher Harmonischer an einer Festkörperoberfläche anstatt in einem Gas untersucht.

Ein weiterer Ansatz ist die Plazierung des Gas-Targets in einem Überhöhungs- resonator. Mit diesem Aufbau wurden Harmonische bis zur 15. Ordnung erzeugt. Abgesehen von einer gesteigerten Effizienz, kann auf diese Weise XUV-Strahlung mit der vollen Wiederholrate des treibenden Oszillators erzeugt werden, d.h. im Bereich einiger zehn bis hundert Megahertz. Hohe Wiederholraten ermöglichen die Verwendung des XUV-Lichts für die hochpräzise Vermessung optischer Frequenzen mittels der Frequenzkammtechnik. Hier wird ein Überhöhungsresonator beschrieben, der innerhalb des Resonators Pulsenergien von mehr als zehn Mikrojoule bei einer Wiederholrate von zehn Megahertz realisiert. Mit dieser hohen Durchschnittsleistung könnten bisher unerschlossene Frequenzen im XUV, wie z.B. der 1S-2S Übergang in einfach geladenem Helium um Größenordnungen genauer als bisher bestimmt werden.

Abstract

The investigation of laser-matter interactions calls for ever shorter pulses as new effects can thus be explored. With laser pulses consisting of only a few cycles of the electric field, the phase of these electric field oscillations becomes important for many applications.

In this thesis ultrafast laser sources are presented that provide few-cycle laser pulses with controlled evolution of the electric field waveform. Firstly, a technique for phase- stabilizing ultra-broadband oscillators is discussed. With a simple setup it improves the reproducibility of the phase by an order of magnitude compared to previously existing methods.

In a further step, such a phase-stabilized oscillator was integrated into a chirped-pulse amplifier. The preservation of phase-stability during amplification is ensured by secondary phase detection. The phase-stabilized intense laser pulses from this system were employed in a series of experiments that studied strong-field phenomena in a time-resolved manner. For instance, the laser-induced tunneling of electrons from atoms was studied on a sub-femtosecond timescale.

Additional evidence for the reproducibility of the electric field waveform of the laser pulses is presented here: individual signatures of the electric field half-cycles were found in photoelectron spectra from above-threshold ionization.

Frequency conversion of intense laser pulses by high-order harmonic generation is a common way of producing coherent light in the extreme ultraviolet (XUV) spectral region. Many attempts have been made to increase the low efficiency of this nonlinear process, e.g. by quasi phase-matching. Here, high-harmonic generation from solid surfaces under grazing incidence instead from a gas target is studied as higher efficiencies are expected in this configuration.

Another approach to increasing the efficiency of high-harmonic generation is the placing of the gas target in an enhancement resonator. Additionally, the production of XUV photons happens at the full repetition rate of the seeding laser, i.e. in the region of several tens to hundreds of megahertz. This high repetition rate enables the use of the XUV light for high-precision optical frequency metrology with the frequency comb technique. With such an arrangement, harmonics up to 15th order were produced. A build-up cavity that stacks femtosecond laser pulses in a coherent manner to produce intra-cavity pulse energies of more than ten microjoules at a repetition rate of ten megahertz is presented here.

With this high average power measuring hitherto uninvestigated optical transition frequencies in the XUV, such as the 1S-2S transition in singly charged helium ions may become a reality.

“La science, mon garçon, est faite d’erreurs, mais d’erreurs qu’il est bon de commettre, car elles mènent peu à peu à la vérité.”

(J. Verne, Voyage au centre de la terre)

Chapter 1

Introduction

Since its discovery, laser light has been used for the investigation of fundamental processes in matter. A plethora of light-matter interaction effects has been examined and exploited since then, from simple macroscopic ones to intricate microscopic ones. Prominent examples of the former are ablation, optical breakdown, and the formation of plasma, usually at the surface of a solid, or the alteration of the refractive index inside it. More complex effects can be studied when entering the microcosm. When atoms are exposed to laser light, the absorption of multiple photons may occur or valence electrons may be excited to Rydberg energy levels. With increasing laser intensity effects like field ionization and related phenomena such as non-sequential double ionization, above-threshold ionization, and the generation of high-order harmonics become important. Triggering these multiphoton processes demands laser intensities where the electric field strengths are comparable to the nuclear fields that bound the electrons to the ionic core. Over the last years, table-top systems have become available that can provide these peak intensities of typically 10[13] to 10[15] W/cm[2] with femtosecond laser pulses. These ultrashort pulses have the advantage of being able to steer the process in a controlled manner and even set off new effects that are not observable with longer pulses. For instance, a longer pulse will saturate the ionization yield well before the pulse maximum. In contrast, a pulse lasting only a few femtoseconds will reach the ionization threshold only close to its peak, thus the available intensities are fully exploited. This is due to the fact that these ultrashort pulses consist only of a few oscillations of the electric field, hence their name few-cycle pulses. At the frequently used wavelengths in the near-infrared, e.g 800 nm for Ti:sapphire systems, one oscillation of the electric field lasts approximately 2.7 fs. Thus, sub-5 fs pulses at these wavelengths are less than two cycles long.

In contrast to the macroscopic effects, the laser-induced microscopic processes (e.g. above-threshold ionization) depend not only on the envelope of the pulse, but rather on the electric field underneath. However, the half-cycles of the electric field in an ultrashort pulse will fluctuate in position (and thus amplitude) from one pulse to the next in a pulse train due to environmental perturbations acting on the laser oscillator. These fluctuations can be characterized by the change of the phase between the pulse envelope peak and the global maximum of the electric field in this pulse, the carrier- envelope phase (CE phase). Hence, in order to fully control the multiphoton processes, intense laser pulses with stabilized electric field waveform are necessary. Over the last years, the joint efforts of the fields of laser stabilization and femtosecond lasers has resulted in the generation of few-cycle pulses with controlled evolution of the electric field.

1. Introduction

The spectra of the ions, electrons, or photons generated by the aforementioned processes reveal insights to the properties of atoms and molecules. Yet, spectroscopy in its conventional form only provides information on the dynamics of fundamental phenomena on the atomic or molecular level after a-priori assumptions. In contrast, repeating the measurement over time results in a direct time-resolved observation of these rapidly evolving systems. Some fundamental aspects and experiments of time-resolved spectroscopy are introduced in chapter 1.1.

Femtosecond laser pulses can be employed for probing and manipulation of the dynamic evolution of a system on a sub-picosecond timescale. Changing the point of view from the time domain to the frequency domain, the global energy-level structure of an atomic system may be investigated with the broad spectral bandwidth associated with femtosecond laser pulses. Information on the separation of atomic energy levels or their fine-splitting is imprinted in the transition frequencies between them. Hence, high-precision measurements of optical transition frequencies can provide insights to the composition of matter, e.g. test the validity of predictions from quantum electrodynamics. Laser oscillators with a controlled CE phase have a spectrum that contains millions of narrow lines. These lines constitute the so-called frequency comb, a powerful tool for measurements of optical frequencies with high precision. The basic concept of optical frequency metrology with frequency combs is discussed in chapter 1.2.

During the course of this work several ultrafast laser sources for time-resolved spectroscopy and optical frequency metrology were conceptualized and constructed. The resultant publications are listed in refs. [1-7].

Three femtosecond oscillators were designed and/or set up, with varying purposes: as stand-alone system, for seeding a multi-pass amplifier system and as a front-end for a parametric amplifier. Chapters 2.1 and 2.2 present the details of these systems. Intertwined with the construction of the broadband oscillators was the development and implementation of a novel CE phase stabilization scheme which resulted in improved control over the frequency comb, verified by thorough characterization (see chapt. 2.3 and refs. [1, 2]).

One of the aforementioned phase-stabilized oscillators was integrated in a compact chirped pulse amplifier. Subsequently, a secondary phase-stabilization loop after the amplification stage was set up to obtain CE phase-stable amplified pulses (cf. chapt. 3.1 and ref. [3]) and a study on the effects from pulse picking in the amplifier on the CE phase was carried out (cf. chapt. 3.1.1 and ref. [4]).

Several phase-sensitive experiments were conducted with this amplifier system in the meantime. In the first, the electron localization in the laser-induced dissociation of deuterium was studied [5]. With a technique similar to the previous setup, signatures of half-cycle cut-offs were observed in the spectra of photoelectrons from abovethreshold ionization (see chapt. 5.3). In another experiment, the light-induced tunneling of electrons from atoms was studied on a sub-femtosecond timescale [6]. With a similar amplifier system, first experiments pertaining to the generation of highorder harmonics from surfaces were performed (cf. chapt. 5.2.1).

As a way to amplify femtosecond pulses at megahertz repetition rates, enhancement cavities (operating at 10 MHz and 1 GHz repetition rate) were designed and/or set up in close collaboration with the group of Prof. Hänsch (cf. chapt. 4.2). Conversion of intra-cavity light to the XUV region was demonstrated with a similar system, thus providing a frequency comb in this spectral region (cf. chapt. 5.2.2 and ref. [7]).

This thesis is structured as follows: the remainder of chapter 1 gives a brief introduction to the research fields motivating the construction of the laser sources. Chapters 2 through 4 present the different ultrafast laser systems, while chapter 5 discusses several applications of these sources.

1.1 Time-resolved spectroscopy

Studying processes inside an atomic or molecular system requires a ‘camera’ with a shutter speed much faster than the timescale of the processes under study. The briefness of the femtosecond laser pulses makes them an ideal tool for the investigation of light-induced processes in a time-resolved manner. A concept for performing time-resolved spectroscopy is the pump-probe configuration. In this setup, a laser pulse (‘pump’) drives the atomic or molecular system to an excited state. A subsequent second pulse probes the temporal evolution of the system. Information on the evolution is extracted from the probe pulse or other products of the interaction, such as electrons or ions.

In this manner, the motion of atoms in molecules was observed and controlled. Laser pulses of ~100 fs duration were employed to study the breaking and forming of chemical bonds, i.e. the occurrence of chemical reactions, in real time. This laid the foundations of the research field femtochemistry, whose key proponent A. Zewail was awarded the Nobel Prize in chemistry in 1999 [8].

Penetrating deeper into matter, the time-resolved observation of electronic motion in atoms necessitates ever shorter pulses, as these processes occur on the timescale of a few femtoseconds down to attoseconds, with the atomic unit of time being 24 as. With the optical period of near-infrared laser light being longer than a femtosecond and the fact that a pulse must contain at least one cycle, only a higher carrier frequency can reach these pulse durations. The nonlinear conversion of femtosecond laser pulses to the XUV spectral region by high-order harmonic generation can produce light bursts that only last a few hundreds of attoseconds [9]. Due to the collinear and perfectly synchronized emission, the generated XUV pulses are together with the fundamental pulses ideal tools for pump-probe experiments. As high harmonic generation is driven by the periodic half-cycles of the fundamental pulse, a train of XUV pulses is emitted. If the CE phase of the pulses is controlled, it can be arranged that only a single half- cycle contributes to XUV photons at the maximum attainable energy and thus, with spectral filtering, a single attosecond pulse is produced [10]. In 2003, the unison of ultrafast laser science and frequency domain laser stabilization has enabled the generation of intense few-cycle pulses with a controlled electric field waveform [11] and hence isolated attosecond pulses. Employing these isolated attosecond pulses in a pump-probe experiment has recently enabled the observation of tunnel ionization dynamics in real time [6]. Laser-induced tunneling occurs near the peaks of the electric field in a pulse where the binding potential is temporarily suppressed. In the experiment, the attosecond XUV (pump) pulse excites atoms of a noble gas to a shake-up state, while the synchronized fundamental near-infrared (probe) pulse ionizes further to produce doubly-charged ions. When scanning the delay between the fundamental few-cycle pulse and the XUV pulse, it was found that tunneling depletes atomic bound states in sharp steps that last ~380 as.

The starting point for generating intense few-cycle pulses with uniform CE phase is a phase-stabilized laser oscillator. However, after the first demonstration of a method for the stabilization of the CE phase of a mode-locked laser in 1999 [12, 13] it was doubted that these nanojoule pulses could be amplified without corrupting the stability of the CE phase. The reasoning was that the amplification process (and the accompanying temporal stretching and compression) would introduce a large amount of phase noise. Nonetheless, with actively compensating for the phase noise introduced in the amplifier, the control over the CE phase could be preserved [14].

With these amplified phase-controlled few-cycle pulses, isolated pulses of 250 as duration were generated at photon energies of ~92 eV [15]. Also, 130 as pulses were produced at an energy of ~36 eV [16] with polarization gating. Access to a wide range of electronic dynamics in atomic core shells requires shorter and more energetic XUV/soft-X-ray pulses. As the frequency conversion by high-harmonic generation is a highly nonlinear process, it depends strongly on the peak amplitude of the half-cycles in the driving laser pulse. Consequently, CE phase jitter creates satellites pulse and introduces timing and amplitude jitter to the isolated attosecond pulse, diminishing its usefulness in pump-probe experiments. Therefore, new methods for increased control over the CE phase are still being explored to this day, for the phase-stabilization of laser oscillators [1, 17-20], as well as for amplifier systems [21-23].

The availability of intense few-cycle laser pulses whose CE phase is controlled with highest accuracy is one of the major preconditions for the generation of attosecond pulses with a duration approaching the atomic unit of time. A wealth of exciting experiments is made possible with these pulses. For instance, the observation of the quantum beating of the atomic wavepacket created by the superposition of the 1S and 2S state in atomic hydrogen with a time constant of ~400 as is appealing. Further, the delay between the emission of photoelectrons from valence and core states of a solid, estimated to be ~100 as could be probed [24]. Thirdly, the dynamics of the electronelectron correlation in the two-photon double ionization of helium could be studied with attosecond pulses in a coincidence experiment [25].

1.2 Optical frequency metrology with frequency combs

In the frequency domain, the spectrum of a pulse train from a mode-locked laser consists of a large number of evenly spaced narrow lines, the so-called frequency comb. This comb of modes has two degrees of freedom, the spacing between the comb lines and the position of the comb center, represented by the comb offset. Both fluctuate arbitrarily in a free-running system. Though if these two comb parameters are being stabilized, the perfectly equidistant comb lines may be utilized as a ‘ruler’ in frequency space. These can then be employed for the measurement of optical frequencies with high precision as the position of each comb line is known with high accuracy.

A measurement is only as precise as the employed definition of the physical units. Using an example of everyday life, a length measurement can only be accurate if an appropriately calibrated metering rule is utilized. As frequency is merely the inverse of time, high-precision frequency measurements need to take the definition of time into account. The second is internationally defined as the duration of exactly 9 192 631 770 periods of the radiation corresponding to two hyperfine levels of the ground state of cesium. A measurement of optical frequencies must thus be referenced to this time standard at ~9 GHz which leaves a frequency gap of approximately five orders of magnitude to be bridged. Frequency combs can serve this purpose and establish a phase-coherent link between the time reference in the rf region and the optical frequencies at hundreds of terahertz. The great improvement to optical frequency metrology associated with them was recognized by the Nobel Prize in physics in 2005 [26, 27]. Many measurements of transition frequencies of neutral atoms and ions have been performed in this way. A schematic overview of such an experiment is given in chapter 5.1. These measurements have allowed experimental investigation into various aspects of fundamental physics. The most striking examples may be the variation in time of fundamental constants such as the fine-structure constant [28-30] or the verification of quantum electrodynamics, e.g. through the frequency of the 1S-2S two- photon transition of hydrogen [31].

Control over the two comb parameters is of paramount importance for these applications. Yet, one of the parameters, the comb offset (the frequency domain embodiment of the CE phase) is difficult to observe and has long escaped measurement and control. Several years elapsed from the first ideas [32] to the first measurement [33] to the first demonstration of control [12, 13]. As mentioned in the previous chapter, progress in stabilization methods is still ongoing. Increasing the degree of control over the offset frequency allows for improved measurements as smaller jitter equals narrower comb lines and thus can permit shorter averaging times for a desired level of measurement accuracy. Further, better long-term stability of the offset frequency can reduce slow drifts of the comb line positions and thus potentially increase the level of precision.

Extending the frequency comb technique to other spectral regions, e.g. the IR or the XUV is an attractive pursuit as hitherto unexplored transition lines can be measured with high precision. In contrast to the IR [34-36], no lasers exist that emit directly in the XUV region. Again, the nonlinear frequency conversion by high-order harmonic generation comes into play. However, the few-cycle pulses provided by standard laser amplifiers do not contain a usable comb structure due to the comb lines being too dense. As the comb line spacing is inversely proportional to the laser repetition period, a higher repetition rate (several megahertz) is needed to use the comb lines for spectroscopy. As a consequence, intense few-cycle pulses with high repetition rate are required. Enhancement resonators that coherently superpose laser pulses in a passive cavity have recently been shown to provide both [37, 7]. Several experiments have indicated that the comb structure survives the frequency conversion [38, 39]. A frequency comb in the XUV opens up new possibilities for spectroscopy of sharp resonances. As one of the exciting goals, the long series of hydrogen spectroscopic measurements could be perpetuated at higher energies. Helium ions are identical to hydrogen in their electronic structure, but carry a higher nuclear charge. Thus, on the one hand, the gross energy levels are at higher energies and more energetic photons are required to excite them. On the other hand, the corrections to the energy levels due to quantum electrodynamics scale as Z 4 (or even higher orders) with the nuclear charge Z and are therefore relatively stronger. For instance, probing the 1S-2S two-photon transition of singly-charged helium ions near 60 nm is a candidate from which a more stringent test of the validity of quantum electrodynamic predictions is expected.

Chapter 2 Ultra-broadband oscillators

Shortly after the invention of the laser in 1960 [40], the first ultrashort laser pulses were generated, already in the picosecond regime [41]. In those days, the race for ever shorter pulse widths began [42, 43]. Figure 1 shows the improvements in reported pulse duration over the last decades. Until the end of the 1980s, ultrashort laser pulses were mainly generated by dye lasers, where a record of 6 fs was achieved in 1987 [44]. Ti:sapphire lasers quickly reached and surpassed this landmark and as a consequence became the preferred devices for ultrashort laser pulse generation. Nowadays, 5 fs pulses that can be compressed externally down to 2.8 fs have been achieved [45-48].

Two different modes of pulsed laser operation have to be distinguished: Q-switching and mode-locking. Q-switching is based on a variable attenuator (e.g. an acousto-optic modulator) inside the cavity that switches from high cavity loss (low Q) to low cavity loss (high Q) after population inversion is created in the gain medium, thereby forming an intense laser pulse [49]. Compared to mode-locking, Q-switching generally results in lower repetition frequencies, higher pulse energies and longer pulse durations (typically nanoseconds). Furthermore, there is no phase correlation between subsequent pulses.

The mode-locking process on the other hand generates ultrashort pulses by establishing a fixed phase relationship between all of the lasing longitudinal modes (fig. 2a). It requires the introduction of an advantage for short-pulse operation over cw¹ operation of the laser. For passive mode-locking in its simplest form, this can be the saturation behavior of a dye or a semiconductor material [50]. A saturable absorber displays a reduced absorption for high intensities compared to low intensities. Thus, a short pulse produces a loss modulation because the high intensity at the peak of the pulse saturates the absorber more strongly than its low intensity wings. In effect, the circulating pulse saturates the laser gain to a level that is just sufficient to compensate for the losses from the pulse itself, while any other circulating low-intensity light experiences more loss than gain and thus dies out during the ensuing cavity round- trips. Hence, self-amplitude modulation of the light inside the cavity occurs. Figure 2b shows the (passive) mode-locking process schematically.

Ideally, mode-locking starts from normal noise fluctuations in the laser, however usually an external perturbation to increase laser noise by mechanical ‘shaking’ of an optical component (e.g. a cavity end mirror) is necessary.

illustration not visible in this excerpt

Figure 1. Improvements in ultrashort pulse generation since the first demonstration of a laser in 1960. Until the end of the 1980s, ultrashort pulse generation was dominated by dye lasers, and pulses as short as 27 fs were achieved. External pulse compression ultimately resulted in pulses as short as 6 fs. Today, Ti:sapphire lasers can generate pulses with only two optical cycles at FWHM². External compression resulted in pulses as short as 2.8 fs. Triangles (squares) denote results from dye (Ti:sapphire) lasers. The filled symbols indicate results directly achieved from a laser; open symbols indicate results achieved with additional external pulse compression.

Mode-locking produces a stable train of ultrashort pulses, though under certain conditions (e.g. high intra-cavity pulse energy), Q-switching instabilities can lead to Q-switched mode-locking, resulting in an unwanted amplitude modulation of the pulse train.

These days, mode-locking in Ti:sapphire lasers is typically accomplished by Kerr-lens mode-locking [51, 52]. It makes use of the intensity dependence of the index of refraction Δn = n2 I(r,t), where n2 is the nonlinear refractive index and I(r,t) the radial- and time-dependent intensity of a short-pulsed laser beam. By placing a Kerr-active material (e.g. the gain medium) at an intra-cavity focus, this results in a Kerr lens that is strongest at the pulse intensity peak. In combination with an aperture inside the cavity, this forms an effective saturable absorber (fig. 3). Usually, a soft aperture configuration is employed, where the reduced mode area in the gain medium improves the overlap with the pump beam and therefore the effective gain.

Currently, Kerr-lens mode-locked Ti:sapphire lasers are producing the shortest fewcycle pulses, consisting of less than two cycles at FWHM at the central wavelength of approximately 800 nm. In order to support such ultrashort pulse widths, obviously a very large spectral bandwidth is necessary.

illustration not visible in this excerpt

Figure 2. (a) Schematic of the pulse train being generated by phase-locking the simultaneously oscillating cavity modes. The forming pulse train is shown with 1, 5, and 40 modes contributing. The mode spacing is 1/Tr = 70 MHz. (b) Principle of passive mode- locking. A saturable absorber is used to obtain self-amplitude modulation of the cavity light. The absorber losses are high for low intensities, but significantly smaller for the high intensities in a short pulse. Hence, a short laser pulse produces a loss modulation with rapid initial loss saturation and a slower recovery, depending on the actual absorber mechanism. The circulating pulse saturates the laser gain so that it just compensates for the losses from the pulse itself. Any other low-intensity light experiences more loss than gain and dies out accordingly.

A constantly widening range of applications in science and industry makes use of femtosecond pulses from Kerr-lens mode-locked Ti:sapphire lasers. A few exemplary areas shall be mentioned here:

In medical applications, ultrashort laser pulses have been used to improve surgical cutting precision, particularly in corneal surgery [53] and brain tumor removal [54]. Using femtosecond pulses reduces secondary damage effects, such as shock waves and cavitation bubbles in tissues [55], as the fluence threshold for optical breakdown decreases with pulse duration.

Telecom applications are mainly the focus when femtosecond pulses are used for the fabrication of optical waveguides and other optical elements, such as splitters, couplers or Mach-Zehnder-interferometers [56, 57]. This micromachining process in fused silica holds the promise of manufacturing truly 3-dimensional integrated photonics elements.

The broad spectral bandwidth associated with ultrashort laser pulses is put to use in optical coherence tomography, a non-invasive interferometric tomographic imaging technique [58]. Having found widespread dissemination in ophthalmology and other biomedical applications, it offers sub-micrometer resolution combined with penetration depths of a few millimeters in tissue. The general scheme is a Michelson- type interferometer with a scanning sample and a reference arm. The short coherence length of the broadband spectrum effects the high spatial resolution, as light outside the coherence length will not interfere.

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Figure 3. Principle of Kerr-lens mode-locking. A Kerr lens is obtained at the focus in the gain material, where the refractive index increases with the incident intensity I(t) by means of the nonlinear refractive index n2. The aperture that effectively forms the fast saturable absorber is commonly implemented as a soft aperture, where the reduced mode area in the gain medium improves the overlap with the focused pump beam, and therefore the effective gain, for the peak of the pulse.

Furthermore, ultrashort oscillators are also employed as seed lasers for laser amplifier systems of all kinds. Two options for amplification of few-cycle pulses are presented in chapters 3 and 4. Ultrashort amplified pulses find application in a variety of areas, of which two are discussed in chapter 5.

2.1 Few-cycle Kerr-lens mode-locked Ti:sapphire oscillator

In this chapter the general setup and the peculiarities of a broadband Kerr-lens modelocked Ti:sapphire oscillator will be presented.

The Ti:sapphire oscillator consists of a linear cavity in an X-folded geometry [59], as shown in figure 4. One of the end mirrors (OC in fig. 4) of this cavity is used to couple out a fraction of the intra-cavity light. The Ti:sapphire crystal (Ti:Al2O3) is optically pumped at 532 nm by a frequency-doubled Nd:YVO4 laser. The pump light is focused by a lens to create efficient population inversion in the crystal. In order to reduce Fresnel reflection losses from its surfaces, the crystal is cut and positioned at Brewster angle. The ensuing astigmatism of the infrared laser beam may be compensated by a suitably-chosen angle of the mirrors used for focusing into the crystal (M3 and M4 in fig. 4). The astigmatism is balanced for

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Figure 4. The schematic of the prism-less Ti:sapphire laser. Pump: frequency-doubled diodepumped Nd:YVO4 laser at 532 nm (Verdi, Coherent Inc.); CP: fused silica compensating plate; L: incoupling lens; M1-5: chirped mirrors; M3, M4: 50 mm ROC³ folding mirrors; OC: output coupler; Ti:S: Ti:sapphire (Ti:Al2O3) crystal; W: fused silica wedged plate

where n = 1.76 is the refractive index of sapphire at 800 nm wavelength, f the focal length of mirrors M3 and M4, the folding angle and t the thickness of the crystal [60].

However, in practice it is found that a slight detuning of the folding angles to smaller values results in better spatial mode quality and easier initial mode-locking, whereas adjusting the angle to marginally larger values increases the output power and vice versa.

The distance between focusing mirrors M3 and M4 has to be close to 2f+ , where is the stability parameter. The range of values of for which the resonator is stable is divided in two separate regions (stability zones). Kerr-lens mode-locking occurs at the border of one of the stability zones. The width of the gap between them depends on the cavity asymmetry

illustration not visible in this excerpt

where d1 and d2 are the length of the long (mirrors M4 - OC) and short (M1 - M3) cavity arm, respectively. Larger represents a larger gap between the zones. Empirically, it was found that a cavity asymmetry of ≈ 2, i.e. a length ratio of the cavity arms of around 2:1 provides the best conditions for mode-locking operation.

Apart from the gain bandwidth of the Ti:sapphire crystal and the reflectance bandwidth of the cavity mirrors, the generation of the broadband spectrum necessary for ultrashort pulses is limited by dispersion inside the cavity. Dispersion, i.e. a wavelength-dependent index of refraction n( ) shifts the longitudinal cavity modes from their equally spaced pattern in the frequency domain. This prevents the mode-locking process from forming an ultrashort pulse. However, if a steady state pulsed operation exists, the pulse looks the same after each round-trip. This effectively dispersion-free propagation is enabled by a delicate interplay between the positive dispersion from the optical elements and several nonlinearities. The dominant nonlinearities in a Kerr-lens mode-locked oscillator are the gain of the laser medium, self-phase modulation and self-amplitude modulation. This balance between the pulse- lengthening effect of the dispersion and the nonlinearities ultimately limits the pulse duration. The pulse formed in the cavity under these circumstances is called a dispersion-managed soliton. The theoretical model predicts Gaussian or super- Gaussian spectral shapes [61], in contrast to secant hyperbolic for pure solitonic propagation. However, measured spectra of ultrashort pulses exhibit complex multi- peaked structures, caused by small imperfections in dispersion compensation.

Intra-cavity dispersion compensation can be achieved either by chirped mirrors or a prism compressor, providing negative group delay dispersion. Design considerations and other aspects of chirped mirrors are presented in detail in chapter 2.2. A prism setup that provides negative second order dispersion is shown in figure 5 [62]. A first prism disperses the beam spectrally. The second prism makes the different colors propagate collinearly. A retroreflector reverses the spatial dispersion. However, the red part of the spectrum travels a longer optical path length compared to the blue part, thus negative dispersion is realized. Prism-based dispersion compensation has the disadvantage of being highly sensitive to cavity alignment, but with the advantage of being continuously tunable in contrast to all-chirped-mirror-based systems. Furthermore, the prism setup introduces large negative third-order dispersion that unfavorably affects the pulse duration. Even with Brewster-cut prisms, a prism compressor has considerably higher overall losses than a set of chirped mirrors.

Ti:sapphire has an extremely large gain bandwidth of more than 300 nm. The Fourier transform of this spectrum already yields sub 4-fs pulse duration. Yet, additional frequency components can be generated inside the cavity by means of self-phase modulation (SPM), which is merely a time-domain manifestation of the Kerr effect.

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Figure 5. Sketch of a prism compressor. This setup introduces negative dispersion, as the longer wavelength components travel a longer distance in the second prism. However the magnitude and sign of the second-order dispersion is strongly dependent on the geometry.

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Figure 6. The longitudinal Kerr effect modulates the refractive index depending on the laser pulse intensity due to the nonlinear part of the refractive index n2. For n2>0 it delays the central part of the pulse with respect to the lower-intensity wings. This causes a red shift in the leading part and a blue shift in the trailing part of the pulse.

While the spectral bandwidth sets the lower attainable limit on the pulse duration, the actual pulse duration also depends on the spectral phase of the pulse. With appropriate compensation of the arising phase distortions, this can result in shorter pulse duration. SPM applies a phase modulation, due to the nonlinear contribution of n2 to the refractive index, to the pulse with an amplitude proportional to the peak power of the pulse. This causes, for n2>0, a red shift at the leading edge and a blue shift at the trailing edge (fig. 6). This generates new spectral components, even outside the gain region of Ti:sapphire, leading to an overall spectral broadening of the pulse. The nonlinearity of either a sapphire plate in a tight secondary cavity focus [45] or of the Ti:sapphire gain medium itself can provide the SPM.

In this manner, several ultra-broadband oscillators spanning more than one octave⁴ were demonstrated [45, 63-66].

Self-phase modulation, along with the Kerr-lens effect and dispersive cavity elements, can bring about several problems in broadband laser oscillators. Firstly, the generated pulses may have a very complicated spectral structure, with strong distortions [45, 67]. These pulses are not easily compressed, external to the cavity, e. g. by a prism compressor. The smoothness of the spectrum is also of concern for applications such as optical coherence tomography, as spectral structures would compromise the image quality through artifacts.

Furthermore, no simple analytical function can be used to model these pulses, therefore their autocorrelation measurement becomes challenging and other methods such as spectral interferometry for direct electric field reconstruction [68] or frequency-resolved optical gating [69] have to be employed.

Additionally, strong SPM action in the laser cavity has the drawback of generating a spectrally-dependent spatial output mode profile, as different spectral components experience a different amount of phase modulation [70, 71]. For the modes that are resonant in the cavity, i.e. that are within the reflectivity bandwidth of the mirrors, the cavity geometry determines the waist size and divergence at the output coupler. Gaussian beam propagation theory predicts a spot size dependence w ~ [1]/[2] (ref. [70]). The beam characteristics of the non-resonant modes, however, can be directly influenced by the Kerr-lensing in the crystal without the self-correcting effects of the cavity.

Several Ti:sapphire all-chirped-mirror laser oscillators were constructed in the course of this work, mainly with the purpose of seeding multi-pass or parametric amplifier systems (see chapt. 3). Requirements due to synchronization to the amplifier pump laser (for the parametric amplifier) or simply space constraints determined the repetition frequency to be in the region between 70 and 80 MHz. The energy of the generated pulses is approximately 5 nJ, with a pump power of 5 to 6 W from a frequency-doubled Nd:YVO4 laser (Verdi V6, Coherent Inc.). This depends on the losses from the mirrors, i.e. quality of the mirror coating. The thickness of the Ti:sapphire crystal is t = 2.6 mm and its absorption coefficient = 5.0 cm−[1] at 532 nm. The radius of curvature of the concave mirrors (M3 and M4 in fig. 4) is r = −50 mm. The transmission of the output coupler is 13%. Specifically designed negatively chirped mirrors (Leybold Optics GmbH) are used to control the cavity dispersion (see chapter 2.2). A pair of thin fused-silica wedges is inserted at Brewster angle for fine- tuning the intra-cavity dispersion.

The spatial profile of the generated beam is approximately Gaussian, however due to the ultra-broad spectrum, the spatial mode profile is spectrally-dependent, as mentioned above. Figure 7 shows the spectral variations in the tangential and sagittal spot size, measured in the far-field 3.75 m from the output coupler.

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Figure 7. Measured beam spot sizes in the far-field 3.75 m from the output coupler as a function of wavelength. Filled (empty) squares denotes tangential (sagittal) plane. The solid and dashed lines are respective fits to the expected [1]/[2] diffraction limit divergence in the region of 700 to 900 nm.

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Figure 8. Spectrum from the oscillator measured with an optical spectrum analyzer (AQ-6315A, Ando Corp.). (a) shows the spectrum on a linear scale, including the gain profile of Ti:sapphire. (b) shows the spectrum on a logarithmic scale, including the sum of GDD contributions in the cavity (3.8 m of air, 2.6 mm of Ti:sapphire, a pair of thin glass wedges, 5 reflections off chirped mirrors (Leybold Optics GmbH)) and a 75 nm adjacent averaging (red solid line).

A typical oscillator spectrum is shown in figure 8 together with the net cavity dispersion. Although the gain bandwidth of the Ti:sapphire crystal extends only from 670 - 980 nm (at -10 dB below the maximum), the generated spectrum from the oscillator extends beyond this region. This means that most of the light at the wings of the spectrum is not generated through lasing but by self-phase modulation in the crystal, a phenomenon previously observed [64, 67].

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Figure 9. Interferometric autocorrelation of the oscillator pulses, measured with an autocorrelator designed for femtosecond pulses (Femtometer, Femtolasers GmbH). The frequency-doubling crystal used is a 10-µm-thick β-BaB2O4 (BBO) crystal. The FWHM pulse duration is 6.2 fs. The dashed line shows the transform-limited pulse as derived from the spectrum in fig. 8 with a FWHM width of 4.7 fs.

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Figure 10. Pulse characterization by spectral phase interferometry for direct electric-field reconstruction (SPIDER). The SPIDER spectrum is recorded by a spectrograph (MS125, Oriel Instruments Corp.), employing a cooled CCD camera (DV420 OE, Andor Technology plc). (a) shows the temporal profile of the pulse. The FWHM of the intensity envelope is estimated as 6.3 fs, in accordance with the result of the interferometric autocorrelation (fig. 9). (b) shows the group delay and spectrum.

The spectrum spans one octave at about -30 dB below the average power level in the 800 nm spectral region.

The sharp roll-off at the short wavelength end and the strong modulation in the long wavelength region clearly do not conform to the theoretical Gaussian spectrum for soliton-like pulses. This is due to imperfect intra-cavity dispersion compensation, especially for higher-order dispersion.

After extra-cavity pulse compression by multiple reflections off of a pair of chirped mirrors (compensating the output coupler’s material dispersion), the temporal characteristics of the output pulses are measured using an interferometric autocorrelator designed for ultrashort pulse diagnostics. Figure 9 shows the measured interferometric autocorrelation trace. The FWHM of the pulse intensity envelope has been evaluated as 6.2 fs.

However, as the autocorrelation technique fails to provide reliable data in the sub-10-fs regime due to the ambiguity in temporal pulse shape, pulse characterization by spectral phase interferometry for direct electric-field reconstruction (SPIDER) [68] was conducted. Temporal profile, retrieved phase and group delay are shown in figure 10. The FWHM of the intensity envelope in the time domain is estimated as 6.3 fs, in accordance with the result of the interferometric autocorrelation.

As is evident from figure 10, satellite pulses are present. However, only a fraction of 14% of the pulse energy is located in these leading and trailing pulses.

Deviation from the ideal case of the transform-limited pulse is mainly attributed to residual cubic dispersion of the cavity dispersion (see also next chapter) at short wavelengths. Its suppression with an improved mirror design will reduce the fractional energy of satellite pulses and compress the main pulse.

2.2 Chirped mirror technology for dispersion control

Chirped mirrors constitute a vital ingredient of broadband oscillators as the key tool for intra-cavity dispersion management. They are also employed for the same reason in enhancement resonators (see chapter 4). Moreover, they are employed in laser amplifiers for third-order dispersion compensation and for pulse compression external to a cavity (cf. chapter 3).

Chirped mirrors were first demonstrated in 1994 [72] and since then have enabled the generation of more and more broadband spectra and hence, shorter pulses. They can compensate the material dispersion of the crystal and other optical elements inside the laser cavity to yield a net slightly negative dispersion. The balance between negative dispersion and nonlinearities such as the Kerr effect shapes a soliton pulse inside the broadband resonator. Therefore, chirped mirrors that provide precise control over the cavity group delay dispersion (GDD) over a wide spectral range, combined with high reflectivity, are mandatory.

Just like all other dielectric mirrors, chirped mirrors are made up of many layers, sometimes up to 70, of alternating high-index and low-index coating materials. Available coating materials are Nb2O5, TiO2, Ta2O5, and HfO2, combined with SiO2 (n = 1.48 at 500 nm). Nb2O5 is a high-refractive-index material (n = 2.35 at 500 nm) that was found to support the broadest spectra.

If the optical thickness of all layers is chosen equal to B/4, interference of all Fresnel reflections generated by the index discontinuity at the layer interfaces will constructively add up for the Bragg wavelength B. Varying the optical layer thickness along the mirror structure during deposition then results in a dependence of Bragg wavelength B on penetration depth z, as shown in figure 11. Chirping the mirror structure therefore allows for the generation of any desired group delay for a specific wavelength. It is obvious that the Bragg wavelength does not have to be varied linearly with penetration depth, but any single-valued function can be used as the chirp law B(z). In principle, this enables the compensation for material second-order dispersion together with arbitrary higher-order dispersion contributions. The physical layer thicknesses are optimized by means of a genetic algorithm to match a pre-defined GDD target curve.

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Figure 11. Chirped mirror structure schematic. (a) Simple chirped mirror structure. The desired effect of a wavelength-dependent penetration depth is accompanied by strong interference effects, mainly from a spurious reflection at the interface to the ambient medium. (b) Double-chirped mirrors provide impedance matching with an additional antireflection (AR) coating on top of the mirror stack and by a duty-cycle modulation inside the mirror stack.

Layer thicknesses of down to 5 nm occur, which makes production of these mirrors close to the design very challenging as the GDD is highly sensitive to small thickness deviations. Most of the chirped mirrors used in the presented broadband oscillators were manufactured with a magnetron sputtering machine (Helios, Leybold Optics GmbH) [73].

Usually, broadband chirped mirrors have pronounced GDD oscillations in the longer wavelength region, as they effectively constitute a Gires-Tournois interferometer. The combination of the highly reflecting chirped mirror stack and an unwanted reflection, e.g. at the interface between coating stack and ambient medium forms this interferometer. Anti-reflection coating on top of the chirped mirror stack provides only partial remedy to the problem, due to the high requirement on the transmittivity of the anti-reflection coating. These GDD fluctuations can broaden the pulse and lead to energy transfer from the initial pulse to satellite pulses. The amplitude of these ripples determines the amount of energy transferred. In order to compensate for these inevitable ripples and to obtain a smooth net dispersion curve, broadband chirped mirrors are typically used in complementary pairs (fig. 12), such that the dispersion oscillations cancel each other. As the angle of incidence shifts the GDD curve in wavelength and as manufacturing imperfections occur, this approach cannot presently avoid GDD oscillations completely.

Progress in mirror design, manufacturing and characterization over the last years has resulted in chirped mirrors with controlled dispersion (residual GDD fluctuations less than 100 fs[2]) and reflectivity (higher than 94%) in the wavelength range of 450 to 1200 nm, corresponding to a bandwidth of 1.3 octaves [73]. These broadband mirrors were characterized with a white-light interferometer [74]. Future use of a cavity-based method [75, 76] will result in more accurate estimates for GDD and reflectivity and potentially better mirror designs.

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Figure 12. Calculated GDD of a broadband chirped mirror pair. (a) Red and blue lines refer to the GDD of the respective mirrors; the green curve shows the average of both. Increasing GDD oscillations in the long wavelength region are clearly visible. The mirrors consist of a stack of 65 alternating layers of SiO2 and Nb2O5. (b) shows the average mirror reflectivity and the uncompensated group delay in a sample linear cavity consisting of three chirped mirror pairs, 2.6 mm of Ti:sapphire, a pair of fused silica wedges, and ~4 m of air.

Despite these achievements in controlling the GDD, it was found that also higher-order dispersion (third order dispersion, etc.) plays a critical role, and may actually be even more important for the generation of smooth output spectra with minimal spectral fluctuations than a full suppression of GDD fluctuations [67].

The required negative GDD for an ideal dispersion-managed soliton can be derived from [77]

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where Ppeak is the peak power of the soliton, FWHM the pulse width (full width at half maximum) and the SPM coefficient,

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where n2 is the nonlinear index of refraction of the crystal, t its thickness, the center wavelength, and w the focus radius inside the crystal [78]. For a typical SPM coefficient on the order of 0.9 rad/MW, an intra-cavity peak power of approximately 7 MW and a pulse width of around 6 fs, this yields a required GDD of D ~ -75 fs[2]. For the optimum Kerr-lensing in the crystal and therefore the shortest pulses, it was found that an approximately equal distribution of the GDD in both cavity arms is beneficial (fig. 13). Assuming perfect dispersion compensation for the moment, equal GDD distribution results in a transform-limited pulse in the center of the gain crystal and at the resonator end mirrors. This yields the highest peak intensity in the crystal, thus the strongest self-phase modulation, the broadest spectrum and therefore the shortest pulses.

Experimentally, it has been observed that a minimum in the GDD curve at around 795 nm facilitates starting the mode-locking process.

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Figure 13. Second-order dispersion map of the cavity. The positive dispersion of the laser crystal D is balanced by the chirped mirrors. In the middle of the dispersion cells (dotted lines), the pulse is chirp free. Additionally, the pulse is modified by SPM in the crystal.

Also the output coupler plays a crucial role in the generation of ultra-broadband spectra. In combination with pump power, mirror losses and other factors, the out- coupling ratio determines the intra-cavity pulse energy and the available output power. It was found that a transmission of around 13% yields the broadest spectra (fig. 14). The spectral extremes are not resonant in the cavity, as they are transmitted by the output coupler. In particular in the long wavelength range, at 10 and 20 dB below the intensity maximum, corresponding to 1029 and 1115 nm, the transmission is 81% and 99%, respectively. The equivalent transmission ratios in the short wavelength range are 26% (-10 dB) and 37% (-20 dB). As the wings of the spectrum are not produced by stimulated emission in the Ti:sapphire crystal but rather through nonlinear phase modulation, the gain in these wavelength regions is much lower. To fulfill the resonance condition (impedance matching), this requires a lower out-coupling ratio. In order to take this into account, a special output coupler design was developed that has lower transmission in the spectral wings (fig. 14). However, the increased spectral bandwidth is traded in for increased pedestals and satellites to the pulse. The output coupler is slightly wedged to prevent out-of-phase back-reflections into the cavity.

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Figure 14. Transmission design curve of a 13% broadband output coupler (black solid line) and corresponding output spectrum (red solid line). The gain curve of Ti:sapphire (green dashed line) is shown for reference. The blue dotted line shows the transmission design curve for a broadband output coupler. In the spectral wings, a lower transmission ratio (~3%) helps to make these components resonant in the cavity.

2.3 The carrier-envelope phase of a mode-locked oscillator

One of the peculiar properties of a mode-locked laser is the intact phase relationship of the optical carrier wave between subsequent pulses. This justifies the definition of a phase of the carrier with respect to the peak of the pulse envelope, the so-called carrier-envelope phase (CE phase). Without additional measures, the CE phase is not constant, but advances by a random amount from pulse to pulse. Therefore, commonly a pulse-to-pulse CE phase shift Δ that includes noise terms is also defined (fig. 15).

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Figure 15. In the time domain, the relative phase between the carrier and the envelope evolves from pulse-to-pulse by the amount Δ . In an unstabilized laser, Δ contains noise contributions.

Here, the electric field of the pulse train is assumed to be represented by

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where Ê(t) is the envelope function of a single pulse, c the carrier frequency, the time between pulses ( = 1/frep), and 0 a phase offset, so that = m Δ + 0 for pulse m. Obviously, the duration of Ê(t) has to be smaller than . Furthermore, the definition of only makes sense for pulses that are sufficiently short compared to the time for a single optical cycle. Otherwise, if the shot noise uncertainty of the electric field maxima is larger than the difference between adjacent field maxima, the well-defined designation of a ‘pulse peak’ becomes impossible. For nanojoule nearinfrared pulses, this limit is at a pulse duration of ~370 fs.

As equation 5 is periodic with period , it can be represented as a Fourier series. Simple algebra (see appendix A.1) shows that its spectrum consists of discrete frequencies, spaced by the repetition rate frep. These so-called comb lines can be addressed by

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where n is a large integer (n ~ 10[6]) and the carrier-envelope-offset frequency fCEO is defined by

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Figure 16. In the frequency domain, the elements of the frequency comb of a mode-locked pulse train are spaced by frep. The entire comb is offset from integer multiples of frep by an offset frequency fCEO = Δ frep/2π. Without active stabilization, both comb parameters are dynamic quantities, which are sensitive to perturbation of the laser.

Therefore, in the frequency domain, the spectrum of a train of pulses from a mode- locked oscillator consists of millions of narrow comb lines (fig. 16). This frequency comb is thus characterized by two properties: (1) the repetition frequency frep of the pulse train that determines the distance between adjacent comb lines and (2) the carrier-envelope-offset frequency fCEO, which introduces a shift of the general comb position.

Another, equivalent way of describing the frequency comb and its two degrees of freedom is by (1) a central carrier frequency fcarrier and (2) the spacing between the comb lines frep. Here, the comb lines are addressed by fm = m frep + fcarrier, where m is an integer that may become negative. Recent studies [79, 80] indicate that this model may describe reality more accurately, as the fixed point of the comb lines, when changing the pulse energy (see chapt. 2.3.3), is close to the carrier frequency at the comb center.

The exact regularity, i.e. the equidistancy of the comb lines has been verified to a very high degree (better than 10-[16]) [81]. However, without active stabilization measures, both quantities are subject to perturbation of the laser. Environmental perturbations, such as air currents, mechanical vibrations and other noise sources will be reflected in the fluctuation of frep and fCEO, as they result in pointing and other instabilities inside the laser cavity. Such instabilities effectuate a change in the net intra-cavity dispersion and also in amplitude-to-phase conversion inside the laser crystal due to the high intra- cavity peak power [82]. Consequently, also Δ fluctuates from pulse to pulse in a non- stabilized laser and active control of the comb parameters is necessary to make the laser oscillator a source of phase-stable pulses and of a frequency-stable comb. Experimentally it was found that fluctuations of fCEO occur on a slower time scale than those of frep [34, 83].

The CE phase, and its frequency domain manifestation fCEO originate from the difference between group and phase velocity inside the oscillator cavity, implying that there is dispersion in the net index of refraction n. Assuming perfect dispersioncompensation by nonlinear effects inside the resonator yields (see appendix A.2)

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where vp and vg are phase and group velocities (averaged over the resonator roundtrip), respectively, and c the center frequency.

This chapter is subdivided as follows: in 2.3.1 measurement of the two comb parameters is discussed in a general way, whereas 2.3.2 presents the details of a novel detection method for fCEO. Chapter 2.3.3 illustrates how to control the comb parameters, i.e. how to give feedback to the laser oscillator. In 2.3.4 the degree of control over fCEO through the detection method presented before is discussed. Exemplary applications of a phase-stabilized frequency comb are presented in the succeeding chapter.

2.3.1 Measurement of the frequency comb parameters

Based on this discussion it is evident that measurement and control of the comb parameters frep and fCEO is desirable. In this way it is not only possible to position the comb lines in frequency space, but also to gain control over the CE phase and as such the electric field waveform underneath the pulse envelope in the time domain (see fig. 15).

Detection of the repetition frequency, i.e. the comb spacing is straightforward: shining the pulsed laser beam (or parts thereof) onto a fast photodiode will provide the corresponding signal. Given that in the frequency comb errors in frep are scaled by n, which is on the order of ~10[6], it is imperative to detect phase noise on frep with high sensitivity. This is accomplished by using a harmonic of frep for detection and stabilization. As phase noise power scales as the square of the harmonic number, in principle, the higher the harmonic the better the phase sensitivity obtained.

On the other hand, measurement of the carrier-envelope-offset frequency is more intricate. Generally, this is due to the fact that the phase of the oscillating electric field usually remains hidden in measurements, as they only depend on the intensity. Only relative phases between optical fields, e.g. in an interferometer can be measured. Consequently, in optics, the complex electric field amplitude remains unknown.

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Figure 17. The principle of the self-referencing technique for detection of the offset frequency fCEO. A comb line with mode number n at the red wing of the comb, whose frequency is fn, is frequency doubled in a nonlinear device. If the frequency comb covers a full optical octave, a line with the number 2n exists simultaneously at frequency f2n. The beat note between the frequency-doubled line and the comb line at 2n yields the offset frequency fCEO.

It was not before the 1990s when the idea emerged to use the comb as its own reference for determining the offset frequency in an interferometric manner [12, 13, 32]. This self-referencing technique uses frequency-doubled spectral components from the long wavelength (‘red’) side (2 (n frep + fCEO)) and mixes them with spectral components from the short wavelength (‘blue’) side (m frep + fCEO). If the two components overlap spectrally (m = 2 n), optical interference produces a heterodyne beating signal that contains information on fCEO (fig. 17).

The underlying fundamental relationship is

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where fbeat corresponds to the beating signal, and n, m and k = 2 n - m are integers. The detected heterodyne beating signal fbeat contains fCEO pair-wise around multiples of the repetition frequency and at dc. Appropriate electronic bandpass-filtering will extract fCEO itself.

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Figure 18. (a) Continuum generation by air-silica microstructure fiber. Self-phase modulation in the microstructure fiber broadens the output of the laser so that it spans more than one octave (solid line). The dashed line shows the original power spectrum entering the microstructure fiber. The spectra are offset vertically for clarity. (b) Scanning electron micrograph image of the input facet of a microstructure fiber. The structure consists of a 2.6 µm diameter core, centered in a hexagonal lattice.

Obviously, this scheme intrinsically requires a comb spectrum spanning at least one octave. Before the recent demonstration of mode-locked oscillators that directly provide an octave-spanning spectrum, spectral broadening external to the laser cavity was employed.

Self-phase modulation in a highly nonlinear medium, usually an optical fiber was found to produce ultra-broadband spectra, especially when a microstructure fiber (photonic crystal fiber, holey fiber) is used to confine the light to a smaller core area and thereby increasing the intensity and the self-phase modulation effect [84, 85]. Microstructure fibers consist of a small fused silica core (2 - 5 µm diameter) surrounded by a honeycomb structure of air holes (fig. 18b). This results in a waveguide with a very high contrast of index of refraction between core and cladding. Thus, single mode propagation for a very large bandwidth is enabled which increases the interaction length tremendously. The design permits shifting the zero point of the GDD (customarily at ~1.3 µm) to the center of the Ti:sapphire laser spectrum which keeps the pulse from dispersing and again increases the interaction length. By these measures, microstructure fibers maintain high intensities over a long distance, thereby leading to the strong broadening of the spectrum as shown in figure 18a. The output spectrum from these microstructure fibers, however, is characterized by strong third- and higher-order chirp. This chirp stems from additional processes inside the fiber such as cascaded Raman effects, fission of higher-order solitons and possibly other incoherent processes [86]. These cannot be compensated easily [48, 87] and makes further use of the pulses impossible.

Higher-order variations of the self-referencing scheme, employing for instance the frequency-tripled long wavelength end and the frequency-doubled short wavelength end of the spectrum also exist [88]. On the one hand, bandwidth requirements are relaxed (less than one octave is required, e.g. the 2f-to-3f scheme just mentioned needs f and 1.5 f), but on the other hand the complexity of the scheme increases due to the additional nonlinear mixing steps. Typically, also the signal-to-noise ratio of the beating signal will become worse.

As the self-referencing scheme makes use of an interferometer of some sort, it is important in the implementation to pay attention to the spectral, temporal, polarization and spatial overlap of the interfering components. Spectral overlap, as the fundamental prerequisite was mentioned already above. Temporal overlap is typically accomplished by a retarding element, e.g. an adjustable block of glass that provides the group delay in one arm of the interferometer.

A variety of technical implementations of this self-referencing technique for the measurement of the CE phase has emerged over the last couple of years. For instance, the interference between different quantum paths in a semiconductor [20] or in optical poling [17] has been shown to contain information on the CE phase. A straightforward method that makes use of a single nonlinear crystal and provides a high degree of CE phase control will be presented in the next chapter.

At present, the most widely used implementation of the self-referencing technique for measurement of the offset frequency makes use of a so-called f-to2f interferometer [13, 89-91] (fig. 19). Prior to the interferometer, a fraction (typically 50:50) of the light from the output of the laser oscillator is branched off and spectrally broadened to an octave in a microstructure fiber. The continuum is then coupled into a Mach-Zehnder interferometer, where a dichroic beam splitter separates the long wavelength (‘red’) part from the short wavelength (‘blue’) part of the spectrum.

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Figure 19. Schematics of a fiber-based f-to-2f interferometer. L: focusing lens; PCF: photonic crystal (or microstructure) fiber; W: wedges for delay adjustment; SHG: frequency doubling crystal (KNbO3); BS: beam splitter; IF: interference filter; PD: photodiode.

The interferometer arm for the long wavelength light contains a KNbO3 frequency- doubling crystal. Several waveplates in the interferometer rotate the polarization such that second harmonic generation in the crystal and recombination of the beams in the second beamsplitter is optimized. The long wavelength interferometer arm contains adjustable fused silica wedges that control the delay between the two arms and ensure correct temporal overlap. In order to reduce unwanted light at the detection photodiode, the region of spectral overlap is selected by means of an interference filter (or, alternatively, a grating combined with a slit). This improves the signal-to-noise ratio, as the beating signal is contained only in a narrow spectral range.

This scheme, although widely used, has several technical and fundamental drawbacks:

(1) the interferometer is alignment-sensitive, (2) in particular, coupling into the tiny core of the microstructure fiber is a tedious task. Fundamentally, the scheme sets limits to the quality of the offset frequency detection and stabilization, as (3) the fact that a beam branched off from the main beam is used for fCEO detection. Relative phase fluctuations and drifts compromise the achievable offset frequency stability. Hence, any phase jitter accumulating between the laser output and the output of the phase control setup appears in the usable laser output, even if the electronic feedback loop works perfectly. Furthermore, (4) the auxiliary setup for fCEO stabilization introduces excessive dispersion, preventing the broadband output from being compressed to a few-cycle pulse. The microstructure fiber also poses a limit to the phase-locking quality, as (5) the fiber-coupling induces amplitude fluctuations. These amplitude fluctuations are written via amplitude-to-phase conversion, mostly due to the Kerr effect, onto the CE phase (conversion coefficient CAP = 3784 rad/nJ). An intensity fluctuation of 1% for a 6 nJ Ti:sapphire oscillator would already result in a phase shift of 227 rad [92].

2.3.2 CE phase stabilization by difference frequency generation

As an alternative to the microstructure fiber-based f-to-2f interferometer, a simple, yet highly effective scheme for stabilization of the CE phase is presented here. It avoids the downsides of the previously mentioned scheme, as it allows for CE phase stabilization directly in the usable laser output. Thus, the CE phase is controlled directly in the beam that is used for applications. Due to the moderate dispersion of the employed nonlinear medium, the transmitted laser pulses are re-compressible. As a consequence, the full laser power is used for inducing the nonlinear processes, resulting in an enhanced beating signal. Also, as no branching off for phase- stabilization is needed, almost the entire laser power is available for application. Furthermore, it relies on the integration of the interferometer into a single monolithic crystal, thereby obviating complex alignment-sensitive setups and improving the achievable CE phase stability. Improved spatial overlap between the two interfering waves, due to the absence of walk-off effects, results in an increased signal-to-noise ratio of the beating signal at fCEO. The absence of a microstructure fiber avoids instabilities (in amplitude and phase) associated with coupling into its tiny core.

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Figure 20. Principle of fCEO detection: an additional frequency comb, fm = m frep + 2 fCEO (fk = k frep), is created in a nonlinear crystal by SHG, top panel (DFG, bottom panel). The frequency comb of the laser fn = n frep + fCEO is spectrally broadened by SPM. With k, n and m being integers, the generated combs partially overlap to produce a heterodyne beat note at fCEO.

When the peak intensity of the pulses and the nonlinearity in a frequency-mixing crystal are large enough, both nonlinear frequency mixing and self-phase modulation (SPM) occur at the same time. Second-order nonlinear frequency mixing can result in second-harmonic generation (SHG) or difference frequency generation (DFG), depending on the phase-matching in the crystal. Previously, beating at fCEO was observed in thin BBO [91] or ZnO [93] crystals, phase-matched to generate second harmonic light.

[...]

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¹ cw: continuous wave.

² FWHM: full width at half maximum.

³ ROC: radius of curvature. The ROC corresponds to twice the focal length.

⁴ An octave is the interval between a certain frequency and double its frequency.