Pulse duration and energy scaling of femtosecond all-normal dispersion fiber oscillators

In this thesis the dissipative soliton operation of mode-locked femtosecond fiber oscillators is investigated in terms of pulse duration and energy scaling. The influence of the oscillator parameters such as spectral filter bandwidth and resonator dispersion on laser operation is analyzed. The fundamental limitations of dissipative soliton operation are identified and explained. Additionally, novel concepts to expand the pulse parameter range of femtosecond fiber oscillators are demonstrated and discussed. The improved understanding of the dissipative soliton operation is used for pulse duration shortening down to 31 fs and energy up-scaling to 0.5 μJ in femtosecond stepindex fiber oscillators.


Introduction
Ultrafast science has become a very broad scientific field with many promising applications. Initially focused on the generation of ultrashort laser pulses, its research areas and applications expanded into other areas of physics, material-processing, microscopy, metrology, medicine, chemistry, and biology [1][2][3][4][5][6][7][8][9][10]. The fast progress in ultrafast sciences has constantly been driven by and relied upon the development and improvement of new ultrashort pulse laser sources. This development concentrates on two different objectives. On the one hand, it is the further scaling of laser pulse parameters which already resulted in the demonstration of ultrashort pulse lasers with average powers of hundreds of watts, peaks powers in the petawatt range, attosecond pulse durations, and energies of several hundred joules [11][12][13]. On the other hand, it is the improvement of already existing ultrashort pulse laser setups in terms of industrial requirements such as total size (footprint), maintenance, price, efficiency and operation stability in order to export ultrafast science to environments outside of research facilities. Note that the second objective is the major motivation for most of the research carried out in the field of femtosecond fiber lasers [14][15][16].
Femtosecond fiber lasers have several advantages compared to the bulk solid-state lasers in terms of total size, user-friendly operation, and lower price, but they still lag behind in pulse energy and duration. To further expand the application fields of femtosecond fiber lasers, and ultrafast science in general, it is necessary to improve the output pulse parameters of these systems. This has been an active research topic during the last decade [15,16] and it is also the main motivation of this thesis.
The present state of the art in femtosecond fiber laser technology is introduced below. Limitations and challenges are explained and important developments in the Introduction energies of up to several millijoules, and average powers of hundreds of watts [11,17,18]. Significant progress in fiber laser performances during the last decade has been mainly due to the development of novel fiber designs [14][15][16]19]. The use of the double-clad fiber geometry enabled efficient pumping with high-power diodes, resulting in a rapid increase in the average output power of femtosecond fiber lasers.
Recent developments of the photonic-crystal-fiber (PCF) technology allowed for the application of large-mode-area single-mode fibers with reduced nonlinearities that increased peak powers of femtosecond fiber lasers by almost two orders of magnitude.  [16,17,20]. These fiber laser systems with pulse durations above 400 fs are already well developed but can still be improved by further reduction of system complexity, e.g. smaller number of amplifier stages and cheaper components.
The presence of wide gain-bandwidths and increased nonlinear spectral-broadening in femtosecond fiber lasers also allows for the generation of shorter pulse durations than it can be achieved in diode-pumped solid-state lasers. In contrast to bulk crystalbased solid-state oscillators, fiber oscillators can even generate output pulses with a spectral bandwidth exceeding the emission bandwidth of the gain medium [21,22].

State of the art
MOPA fiber laser systems based on chirped pulse amplification (CPA) are able to amplify sub-200 fs pulses up to the energies of several µJ [23]. In parabolic pulse fiber amplifiers strong nonlinear spectral-broadening can be exploited to generate parabolicshaped pulses with energies of several hundreds of nJ and durations below 100 fs [24]. However, many applications, especially in material processing, metrology, and medicine utilize sub-50 fs pulses with energies of several tens of nJ or even µJ and sub-100 fs pulses with energies of up to the millijoule level [1,25]. Fiber laser systems and alternative diode-pumped laser systems can only access this parameter range using nonlinear pulse compression stages at the output end of the laser system, which significantly increase the impact of noise and sensitivity to adjustment. So far no fiber lasers without nonlinear compression stages operating in this parameter range have been demonstrated and therefore more expensive Ti:sapphire lasers must be used.
These applications would significantly benefit from the availability of cheap, userfriendly, and compact fiber laser alternatives.
The most widely used scheme for the generation of femtosecond pulses in fiber lasers is a MOPA setup with CPA [15,16]. A train of chirped femtosecond seed pulses from a mode-locked fiber laser is amplified in one or more power amplifier stages followed by a pulse compression afterwards. In such MOPA schemes the output pulse durations and energies depend mainly on the corresponding parameters of the seed oscillator. Therefore, the above mentioned reduction of system complexity and shortening of the pulse durations of femtosecond fiber lasers can be achieved by an improvement of the characteristic of the seed oscillator. Shorter pulse durations from the seed oscillator result directly in shorter pulse durations after amplification. Whereas, higher output pulse energies reduce the required amount of amplification, the number of amplifier stages, and also the increase in the pulse duration due to gain-narrowing effect. These reasons motivated active research of the scaling properties of mode-locked fiber oscillators during the last decades. The pulse durations and energies of several demonstrated step-index fiber oscillators are presented in Fig. 1.1 (b). So far, pulse durations as short as 28 fs and output pulse energies up to 31 nJ have been demonstrated from the step-index femtosecond fiber oscillators [21,26]. Results of this thesis are shown by the red triangles. Only step-index fiber oscillators are shown in Fig. 1.1 to highlight the influence of the pulse dynamics. Larger pulse energies have also been demonstrated by use of PCFs with larger mode-field-diameters [27][28][29][30].
Mode-locked fiber oscillators allow for large amounts of dispersion and nonlinear-Introduction ity per round-trip that give rise to a variety of different mode-locking regimes such as fundamental, dispersion-managed, and dissipative solitons [31][32][33]. The dynamics of fundamental and dispersion-managed solitons are already well understood and the existing limitations in the achievable pulse energy and duration have been identified. Dissipative soliton operation of mode-locked fiber oscillators has been first experimentally demonstrated in 2006 [34,35]. Since then, this topic is an active area of research in laser science. Dissipative solitons have been generated in ytterbium and erbium fiber oscillators with total normal resonator dispersion. These dissipative solitons rely on a balance between gain and loss per round-trip, group velocity dispersion, self-phase-modulation, saturable absorption, and spectral filtering [33]. They have a large positive chirp throughout the resonator that allows for accumulation of large amounts of nonlinear phase and the generation of output pulse energies of several tens of nanojoules. This corresponds to an increase by more than one order of magnitude in pulse energy compared to fundamental and dispersion-managed solitons. Due to the toleration of a much higher amount of nonlinear phase-shift, the dissipative solitons can also experience stronger nonlinear spectral broadening per round-trip than dispersion-managed solitons and may achieve shorter pulse durations. Therefore, the dissipative solitons generated in all-normal dispersion oscillators are promising candidates for further pulse duration and energy scaling.

Objectives and motivation
The aim of this thesis is to investigate the pulse duration and energy scaling of dissipative solitons in femtosecond fiber oscillators. The dynamics of dissipative solitons have already been studied in several experimental and theoretical publications [33,36,37]. However, until now, possible influences of oscillator parameters such as spectral filter bandwidth, dispersion, and saturable absorption on the output pulse characteristics are not sufficiently understood. Especially, it has already been observed that the resonator dispersion has a strong influence on dissipative soliton dynamics that requires further investigations. For example, numerical simulations predict shorter pulse durations at smaller resonator dispersions, and analytical solutions of the Ginzburg-Landau equation show an increase of pulse energy at larger resonator dispersions [38,39]. Both effects have not been systematically investigated experimentally, and no limitations on pulse duration and energy scaling have been identified. Fur-1.3. Organization of the thesis thermore, no qualitative or intuitive explanations of these effects have been given.
Another uninvestigated effect is the pulse energy limitation in the dissipative soliton regime. In previously reported dissipative soliton oscillators, the pulse energy limitations have been attributed to over-driving of the mode-locking mechanism [40,41].
Numerical simulations that neglect this effect predicted twice as large output pulse energies with appropriate saturable absorber parameters [40,41]. Evidently, further experimental and theoretical investigations are required and may provide access to a new range of pulse parameters from dissipative soliton oscillators.
The first objective of this thesis is to use flexibility in the adjustment of fiber dispersion at the 1550 nm emission wavelength of erbium-doped fiber oscillators to determine the possible limitations on pulse duration of dissipative solitons. Unfortunately, until now, the characteristics of all-normal dispersion ytterbium fiber oscillators, in terms of pulse duration and energy, could not be reproduced in fiber lasers operating at other wavelengths (e.g. erbium, thulium). Thus, it is necessary to first demonstrate that the performance characteristics of dissipative solitons can be transferred to erbium fiber oscillators. The next objective of this thesis is in investigations of laser pulse energies achievable at large resonator dispersion and the influence of over-driving of the mode-locking mechanism.
After the improvement in performance of mode-locked all-fiber integrable oscillators, there are still several applications which cannot be addressed by these laser systems.
For example, µJ laser pulses with sub-50 fs durations cannot be generated by all-fiber lasers. Therefore, an alternative, recently demonstrated fiber oscillator concept will be investigated and its scaling properties in terms of pulse duration and energy will be analyzed. This oscillator concept is based on the combination of a short active fiber section with a free-space resonator. The short active fiber section provides large roundtrip gain and enables the accumulation of large nonlinear phase-shifts, required for the generation of short pulse durations. The free-space resonator supports pulse energies comparable to bulk solid-state oscillators. This oscillator concept allows scaling of the pulse parameters by variation of the active fiber length.
Introduction characterization methods. In chapter 3, main performance characteristics of dissipative soliton ytterbium fiber oscillators are reproduced in an all-normal dispersion erbium fiber oscillator. The adjustment of resonator dispersion of this oscillator is then investigated in chapter 4, resulting in the demonstration of sub-50 fs pulse durations.
In chapter 5, the all-fiber integration of this setup is realized. Chapter 6 is devoted to investigations of limitations in pulse energy scaling by increasing the resonator dispersion. The influence of over-driving of the mode-locking mechanism is determined in chapter 7. A full theoretical and experimental analysis of mode-locked fiber oscillators combining an active fiber section with a free-space resonator is given in chapter 8.
Finally, in chapter 9 the results achieved in this thesis are summarized and an outlook is given.

Fundamentals
Research and development of mode-locked fiber lasers are active and challenging topics in modern science. While, nowadays the experimental realization of mode-locked fiber lasers can be very simple, the physics of optical pulse dynamics is highly sophisticated. Complex physical effects which are still unexplained or can only be understood by substantial amounts of numeric simulations, one can easily observe with apparently trivial setups. Numerous physical models have been developed to describe the physics of mode-locked fiber oscillators but accurate quantitative and even qualitative predictions are still difficult to obtain. This chapter highlights some fundamentals of mode-locked fiber oscillators, beginning with the theory of nonlinear pulse propagation in optical fibers. Taking into account the boundary conditions of the optical resonator and the nonlinear transmission function of the saturable absorber, this theory is used to describe the pulse dynamics of mode-locked fiber oscillators. Next, different mode-locking regimes of fiber oscillators are introduced and the benefits of dissipative soliton operation are explained.
The final part of this chapter comments on the experimental methods required for the set-up and characterization of mode-locked fiber oscillators.

Pulse propagation in optical fibers
In a mode-locked fiber oscillator an optical pulse travels repetitively through the resonator and is reproduced after each round-trip. To understand the physics of modelocked fiber oscillators, it is necessary to discuss the pulse propagation in optical fibers.
This section introduces some standard notations and well-known formulas describing ultrashort laser pulse propagation [42]. Additionally, two solutions of the nonlinear pulse propagation equation, which are important for the understanding of several mode-locking regimes, are presented.

Pulse-propagation equation
An optical pulse is defined as an amplitude modulation of a fast oscillating optical carrier-wave. The electric field E(t, x, y, z) of a laser pulse inside an optical fiber is Fundamentals given by: where z is the propagation distance along the fiber, F (x, y) is the modal distribution of the fiber mode, A(t, z) is the electric field envelope (normalized so that its absolute square gives power), and β(ω 0 ) is the wave number of the fiber mode at the carrierfrequency ω 0 of the electric field. The modal distribution F (x, y) and the wave number β are defined by the fiber geometry and are constant along the fiber length. The temporal and spectral properties of the optical pulse are fully determined by the electric field envelope A(t, z). The evolution of the temporal and spectral pulse profiles during propagation through the optical fiber is described by the nonlinear pulse-propagation equation [42]: where g is the fiber gain, ∆ω is related to the limited gain-bandwidth, β 2 and β 3 are the second-and third-order dispersion values of the fiber, γ is the nonlinear parameter, and T R is the first moment of the nonlinear Raman response function. In Eq. (2.2) a frame of reference moving with the pulse at the group velocity v g is used, which is obtained by the substitution: 3) The first two terms on the right hand side of Eq. (2.2) account for the gain with the limited bandwidth ∆ω inside the optical fiber. The temporal pulse broadening due to fiber dispersion is described by the next two terms. The nonlinear terms incorporate, in order of their appearance, the effects of self-phase-modulation (SPM), self- For the above mentioned reasons, this equation will be used throughout this thesis to describe laser pulse-propagation in optical fibers.

Optical solitons
In case of no gain (g = 0) and anomalous dispersion (β 2 < 0), the NLS equation has a class of exact solutions referred to as optical solitons. These solutions are divided in two groups, the higher-order and fundamental solitons. An infinite number of different higher-order soliton solutions exists. They are characterized by a periodic reproduction of their temporal and spectral pulse profiles during propagation. Higherorder solitons are unstable and almost directly break-up into fundamental solitons as soon as SRS or third-order dispersion is present. Because of this instability, higherorder solitons are usually not utilized in ultrashort pulse lasers. Fundamental solitons, on the other hand, are attractive solutions of the NLS equation. They preserve their temporal and spectral pulse shape during propagation and are able to adapt their pulse parameters to adiabatic changes in pulse energy and fiber properties. If the parameters of any initial pulse are sufficiently close to a fundamental soliton, the initial pulse will reshape to this fundamental soliton during propagation, thereby losing a small part of its energy to a dispersive wave. These properties make fundamental solitons highly useful for long distance data transmission and ultrashort pulse generation.
The electric field envelope of the fundamental soliton is given by: where the peak power A 0 and the pulse duration T 0 are related by the "area theorem": This equation shows that the pulse energy of the fundamental soliton can only be increased by reducing the pulse duration or changing the fiber parameters.

Fundamentals
An important parameter for the fundamental solitons is the soliton period z 0 , given by: The soliton period determines the propagation distance of the fundamental soliton required to readjust itself after small changes in fiber parameters and pulse shape. At higher soliton pulse energies the soliton period becomes shorter which results in less stable pulse propagation. At high peak powers and short pulse durations fundamental solitons are also destabilized by third-order dispersion and SRS, breaking-up into multiple pulses with smaller energies. This soliton fission limits the pulse energies and durations achievable with fundamental solitons in step-index fiber lasers to a few nJ.

Parabolic pulses
Parabolic pulses can be considered as counterpart to fundamental solitons at normal fiber dispersion values (β 2 > 0). They are self-similar solutions to the extended NLS equation Eq. (2.4). Their temporal and spectral pulse forms are preserved during nonlinear pulse propagation while the pulse duration, peak power, and chirp change with the propagation distance [43]. In the presence of gain (g > 0) parabolic pulses are an asymptotic solution of the NLS equation [44,45]. Thus, any initial pulse will evolve to a parabolic pulse during propagation. In contrast to solitons, only one asymptotic parabolic pulse solution exists for a given set of fiber parameters and is given by: where U 0 is the pulse energy at z = 0 and φ 0 is an arbitrary constant. The derivation of this parabolic pulse solution is an important demostration of the method of dimensional analysis and is presented in appendix A.
Due to their self-similar propagation properties and the analogy to fundamental solitons, the parabolic pulses in amplifiers are also referred to as similaritons. Just as fundamental solitons, similaritons have a constant pulse chirp during propagation.
Furthermore, the "area theorem" (see Eq (2.6)) is also valid for similaritons and is

Mode-locked fiber oscillators
given by: (2.9) where A p = A 0 exp(g · z/3) is the peak power of the similariton. The Fourier-transformlimited (FTL) duration at half-maximum of the similariton T 0 and the multiplicative factor are determined numerically from the similariton spectrum [46]: , (2.10) In opposite to fundamental solitons, which have transform limited pulse durations, similaritons have a strong positive pulse chirp and significantly longer pulse durations. The long pulse durations allow for higher pulse energies and the strong chirp prevents pulse break-up due to third-order dispersion and SRS.
The self-similar propagation of parabolic pulses has already been theoretically predicted in 1993 [43]. However, parabolic pulses found applications in ultrafast laser science only after they have been identified as asymptotic solutions of the NLS equation with gain in 2000 [45]. This knowledge enabled to employ nonlinear pulse distortions during amplification for pulse shaping instead of avoiding nonlinearities by use of CPA. Since then, parabolic pulse amplifiers have been used to generate sub-200 fs pulses with energies of hundreds of nJ, independently of the seed oscillator parameters [24].

Mode-locked fiber oscillators
The progress in mode-locked fiber oscillators has been strongly driven by the develop-

Mode-locking
A fiber oscillator is realized by seeding of a fiber amplifier section with a small fraction of its own output. The remaining power is out-coupled from the oscillator and used as the laser output. The fiber oscillator supports laser operation at several longitudinal modes simultaneously. It is referred to as a mode-locked oscillator when these longitudinal modes have a definite phase-relation to each other. In the modelocked oscillator an optical pulse is formed by interference of phase-locked longitudinal modes and propagates repetitively through the resonator, reproducing itself after each round-trip. In practice, mode-locked laser operation is achieved by implementing saturable absorbers with intensity dependent transmission functions into the oscillator. The transmission of a saturable absorber increases at higher intensities of the incident laser light. Therefore, incorporation of a saturable absorber suppresses continuous wave and supports pulsed laser operation [47]. Three different types of saturable absorbers are usually applied in mode-locked fiber oscillators. These are semiconducting saturable absorber mirrors (SESAMS) [48], carbon nanotubes [49], and absorbers exploiting the nonlinear polarization evolution (NPE) in optical fibers [47,50]. Carbon nanotubes and SESAMs are based on the same physical principle, which is the saturable absorption in a bandgap energy-level structure. This energy-level structure results from the geometrical shape of the carbon nan- tained saturable absorber is given by [50]: where α i (i = 1, 2, 3) are the rotation angles of the three wave-plates, L is the fiber length, and A in (t) and A out (t) are the electric field envelopes in front and behind the PBS. To simplify the analysis, in the derivation of Eq. (2.11) the electric field envelope during pulse propagation inside the fiber section is assumed constant.
All oscillator setups presented in this thesis employ a NPE-based saturable absorber with the rejection port of the PBS as the laser output. The NPE mode-locking has been chosen due to its simple and continuously tunable adjustment and its much higher damage threshold compared to carbon nanotubes and SESAMs.

Numerical simulations and analytic analysis
The control of the output pulse parameters of mode-locked fiber oscillators requires theoretical understanding of the pulse dynamics. The pulse parameters of modelocked fiber oscillators can be analyzed either by numerical simulations or analytic models.
To simulate a mode-locked fiber oscillator, the evolution of an initially arbitrary pulse in the fiber section is numerically calculated. The nonlinear pulse-propagation according to the NLS equation with gain (2.4) can be numerically simulated using the symmetrized split-step Fourier method algorithm [42]. This algorithm is based on alternate step-wise application of the fiber dispersion in the spectral domain and the An analytic model which provides qualitative information and design guidelines is the master-equation formalism developed by Haus et al. [47]. Stable mode-locked pulse operation requires that the electric field envelope A(t) reproduces itself after each round-trip, except for a constant phase-shift φ. By averaging all pulse-propagation effects over one round-trip, this condition can be expressed by the master equation: Here, D is the total resonator dispersion and Ω is the full width at half maximum

Pulse dynamics
The main function of the saturable absorber in mode-locked fiber oscillators is to initiate pulsed laser operation. It is important to clarify, that the saturable absorber has  and achieve output pulse energies of a few nJ [21].
At the positions of minimum pulse duration, DM solitons can acquire a non-monotonic chirp, which is characterized by the same instantaneous frequencies at different time moments and leads to the effect of optical wave-breaking, limiting the achievable pulse duration and energy [51]. Additionally, the spectral bandwidths and energies

Mode-locked fiber oscillators
of the DM solitons are limited by the influence of Raman shift, third-order dispersion, and pulse distortions due to excessive self-phase-modulation.

Similaritons
Shortly after the demonstration of parabolic pulse formation in normal dispersion fiber amplifiers (see section 2.1.3), the discovered pulse dynamics have been implemented into mode-locked fiber oscillators [33,41,52]. In the similariton regime the output pulses from a parabolic pulse fiber amplifier are temporally compressed by use of negative dispersion or a narrow spectral filter. Afterwards, the fiber amplifier is from dissipative soliton oscillators [38]. As for similaritons, the limitations in pulse duration and energy have again been attributed to over-driving of the mode-locking mechanisms [40]. Numerical simulations have predicted that pulse energies twice as high and compressed durations as short as 30 fs can be achieved [38]. Furthermore, no fundamental limitations on pulse energy or duration have been identified or predicted. Therefore, dissipative solitons generated in all-normal dispersion oscillators are promising candidates for further pulse duration and energy scaling. All fiber oscillators presented in this thesis operate in the regime of dissipative solitons.

Experimental methods
As explained in the previous section, stable single-pulse mode-locked operation of a fiber oscillator requires an appropriately adjusted saturable absorber, which initi-

Experimental methods
ates pulsed laser operation, and the existence of a single pulse solution to the NLS equation with gain, Eq. (2.4), which fulfills the boundary conditions of the resonator.
Despite the complex pulse dynamics and requirements to the saturable absorber parameters, stable single-pulse operation can be easily achieved in most fiber oscillator setups. However, the experimental identification of stable single-pulse operation and the characterization of the output pulse parameters are more complex. To ensure that no multi-pulse or q-switched operation is present and to determine the pulse parameters, the fiber oscillator output must be completely analyzed in both the optical and the radio-frequency domains.
This section introduces the experimental methods used to characterize the output of mode-locked fiber oscillators. The measurements of the temporal and spectral output pulse properties are explained and methods used to verify stable single-pulse operation are discussed.

Optical pulse characterization
An optical pulse is fully determined when its power and phase distributions, either in the temporal or spectral domain, are known [53]. While the spectral power distribution can be easily measured using chromatic angular dispersion of optical gratings, the determination of the temporal pulse profile has become possible since the demonstration of frequency-resolved-optical-gating (FROG) and is still challenging.
A common method to characterize ultrashort optical pulses is the simultaneous measurement of the optical spectrum and the intensity autocorrelation. Although, the intensity autocorrelation provides rather an estimation than a true measure of pulse duration, its simple adjustment and reproducibility have made it a frequently used measurement tool in the field of ultrashort laser pulses. The intensity autocorrelation measurement uses the optical pulse and its temporally delayed copy for frequencydoubling in a second-harmonic-generation (SHG) crystal. The intensity of the frequencydoubled laser light as a function of the temporal delay is referred to as intensity autocorrelation function. For an optical pulse with the electric field envelope A(t), the intensity autocorrelation function is given by: Pulse duration and width of the autocorrelation function are related by a multiplicative factor that can be derived from the temporal pulse shape. Thus, for a given pulse Fundamentals shape, the pulse duration can be determined by measuring the autocorrelation function. Unfortunately, in most cases the real pulse shape is unknown, and in order to estimate the duration of a measured pulse, a guess of the pulse shape has to be made.
An intuitive and widely used assumption is that a pulse close to the compression limit (less than 20% deviation from the FTL duration) has the same shape as the FTL pulse, calculated from the measured pulse spectrum. This assumption allows to estimate the pulse duration by measuring intensity autocorrelation and spectrum.
An exact measurement of the electric field envelope of an optical pulse can be obtained using FROG [53]. In a FROG measurement the frequency-doubled output of the intensity autocorrelator is spectrally resolved. The resulting two-dimensional FROG trace fully determines the measured optical pulse, except for an ambiguity in the sign of time, and is given by: (2.14) The electric field envelope A(t) must be derived numerically from the measured FROG trace using two-dimensional phase retrieval algorithms. The complexity of the FROG method limits its use in the field of femtosecond fiber lasers. Commercially available FROG devices can very accurately measure the clean pulse shapes generated in modelocked solid-state oscillators. However, the pulses generated in mode-locked fiber oscillators, especially in the dissipative soliton regime, are significantly more complex due to the increased amount of nonlinearity. The nonlinearity can result in fast modulations in the pulse spectrum and in low-amplitude side-structures in the temporal pulse profiles, which extend over the range of several picoseconds. Therefore, the accurate characterization of optical pulses generated in fiber oscillators requires higher spectral resolutions and larger time-spans than that used for solid-state oscillators.
Commercial FROG devices can only be used to estimate the temporal pulse shape and additional measurement of the intensity autocorrelation must be performed to verify the results.

Verification of stable single-pulse operation
Apart from the stable single-pulse mode-locking, fiber oscillators support a variety of other operation states. These operation states are cw and mode-locked q-switching, period-doubled mode-locking, and multi-pulse mode-locking [48,54,55]. To distinguish the undesired operation states from the stable single-pulse mode-locking, the

Experimental methods
laser output must be analyzed in the radio-frequency domain.
In cw q-switched operation the fiber oscillator generates high energy pulses with durations in the microsecond range at repetition rates of a few kHz. The cw q-switched operation can be easily identified by recording the oscillator output power versus time with a combination of fast photodiode and oscilloscope. During mode-locked q-switching the energies of the output pulse train are modulated at kHz frequencies.
This state of operation can also be observed using an oscilloscope. However, this state of operation can be more easily distinguished from the stable single-pulse operation by measurement of the radio-frequency spectrum of the laser output. The stable single-pulse operation results in periodic spikes in the radio-frequency spectrum at multiples of the fundamental repetition rate of the oscillator [56]. During mode-locked q-switching smaller spikes can be observed at a distance of a few kHz from the main spikes. Additionally, the width of the main spikes increases.
In mode-locked fiber oscillators it is also possible that the pulse is reproduced only after two or more round-trips. This is referred to as period-doubling and is identified by the presence of additional spikes in the radio-frequency spectrum at multiples of half the fundamental repetition rate.
To distinguish single-pulse operation from multi-pulse mode-locking the oscilloscope measurement of the output pulse train must be combined with the intensity auto- [59] could demonstrate dissipative solitons from an erbium fiber oscillator, but the achieved pulse energies did not exceed the energy values possible in the DM soliton regime. Ruehl et al. [60] reported an erbium fiber laser with large normal resonator dispersion and pulse energies of 10 nJ. The demonstrated output pulse spectra were strongly distorted by Raman shift with a maximum at 1640 nm and had no similarity with the spectra generated in ANDF ytterbium lasers. Furthermore, only one of the previously reported erbium fiber lasers operating in the dissipative soliton regime made use of a spectral filter inside the resonator [59], although studies of ANDF ytterbium lasers clearly showed the strong influence of a spectral filter on dissipative soliton dynamics [33,36]. This chapter presents an ANDF erbium oscillator operating in the dissipative soliton regime mode-locked by NPE and with a spectral filter inside the resonator. To the best of our knowledge this is the first report of an erbium fiber laser with output pulse characteristics similar to comparable ytterbium-based ANDF 3.1. Experimental setup oscillators (15 nJ and 500 fs in [61]) in terms of pulse energy and duration.

Experimental setup
The ANDF erbium oscillator setup is shown in Fig. 3

Results
The fiber oscillator setup is optimized in terms of output pulse energy by reducing the erbium-doped fiber length while keeping the length of passive fibers constant. Additionally, at shorter erbium-doped fiber lengths the pump power is increased to account for the weaker pump light absorption and higher average output powers. As can be High-power dissipative solitons from an all-normal dispersion erbium fiber oscillator seen in Fig. 3.2, the achievable output pulse energies increase with decreasing erbium fiber length until at 59 cm fiber length the highest observed output pulse energy of 31 nJ is obtained. The output pulse energy of the oscillator is limited by nonlinear effects, which include pulse distortions by Raman shift, self-phase-modulation, and over-driving of the NPE. The oscillator is expected to have a strong output coupling at the NPE-port and only a small fraction of the pulse enters into the 57 m long passive fiber. Since the erbium fiber has additionally a two times smaller mode-field diameter than the passive fiber, it is the major source of the nonlinear effects inside the oscillator.
Therefore, shortening of the erbium fiber leads to the observed increase in achievable output pulse energies. In case of fiber lengths shorter than 59 cm, the available pump power and pump light absorption are not sufficient to achieve mode-locked operation.
Consequently, in the presented oscillator we use the erbium fiber length of 59 cm to achieve the highest possible output pulse energy.  The relatively large difference between the compressed pulse duration and the Fourier- Therefore, ANDF erbium oscillators should allow for the generation of dissipative solitons with shorter pulse durations than those reported from ANDF ytterbium oscillators (operating at 1.03 µm). In the following, the ANDF erbium oscillator of the previous chapter is redesigned in order to achieve a small normal total resonator dispersion. This results in the demonstration of the shortest pulses, reported so far, from an ANDF oscillator.

Theory
To explain the pulse duration shortening at smaller total dispersion values of the oscillator a similar approach as in [47] can be used. This approach results in an analytical relation between pulse duration and total dispersion of an ANDF oscillator.
Stable mode-locked pulse operation requires that the electric field envelope A(t) (normalized such that its absolute square is power) reproduces itself after each round-trip, except for a constant phase-shift φ. By averaging all pulse-propagation effects over one round-trip, this condition can be expressed by the master equation (2.12): To derive the scaling properties of the Fourier-limited pulse duration from Eq. (2.12), we assume a strongly chirped parabolic pulse given by: with a b, where the parameters a and b determine the pulse duration and chirp.
Substituting this pulse in Eq. (2.12) and solving it up to second order in t gives a set of equations from which, with the approximation a b for a strongly chirped pulse, the Fourier-limited pulse duration can be obtained: This relation is identical to the "area-theorem" for amplifier-similaritons in Eq. (2.9).
In the derivation of Eq. (4.2) we assume that the absorber is saturated by the dissipative solitons and set the slope of the transmission function of the saturable absorber equal to zero (α = 0). As can be seen from Eq. (4.2), shorter pulse durations can be achieved by decrease of the resonator dispersion D or increase of the nonlinear phase accumulated over one round-trip δ |A 0 | 2 . However, in ANDF lasers mode-locked by NPE the amount of nonlinear phase-shift is limited by nonlinear effects such as overdriving of the NPE and Raman scattering (see chapter 7). Thus, as has already been mentioned, the total dispersion has to be reduced to achieve shorter pulse durations.
The ANDF oscillator setup presented in the next section has been designed to obtain the smallest resonator dispersion achievable with the available fiber components.

Experiments
The setup of our ANDF erbium oscillator is shown in Fig. 4   The radio-frequency spectrum of the output pulse train is shown in Fig. 4.2 (f). The narrow linewidth and the constant height of the radio-frequency peaks as well as the noise-suppression at 40 dB confirm that no Q-switching, period-doubling, or higherharmonic mode-locking are present. Single pulse operation is verified by the intensity autocorrelation, the oscilloscope trace, the constant height of the radio-frequency peaks.

Discussion
For achieving the shortest possible pulse duration the choice of the filter bandwidth is essential. Although the filter bandwidth does not significantly affect the Fourierlimited pulse duration of our laser, it has an influence on the pulse compressibility.
By using filters with bandwidths of 22 nm and 47 nm, pulses not shorter than 60 fs could be obtained. To explain this effect we note that the pulse compressibility is determined by the magnitude of the higher-order phase terms in the output pulses.
An increase of the higher-order phase terms worsens the pulse compressibility. If a smaller filter bandwidth is used, the pulses must accumulate more nonlinear phase over one round-trip to achieve output pulses with the same spectral bandwidth, and therefore, the higher-order phase terms of the output pulses are increased. If the filter bandwidth is too large the higher-order phase terms at the wings of the pulses are not filtered out and are accumulated over several round-trips, which also leads to an increase of the higher-order phase terms in the output pulses. Thus, at the optimal filter bandwidth the nonlinear phase-shift accumulated over a single round-trip should be sufficiently small to allow for efficient compression, and the spectral filtering should be able to restore a linearly chirped pulse after each round-trip.
It should be possible to further decrease the output pulse duration by using erbiumdoped fibers with lower dispersion and by replacing the Metrocor fiber with a dispersion decreased fiber. Furthermore, erbium fiber absorption at wavelengths below 1480 nm may have limited the spectral bandwidth of the output pulses in our setup.
Using pump light at 980 nm can result in further decrease of the pulse duration.
In summary, we have demonstrated an all-normal dispersion erbium fiber oscillator generating dissipative solitons that have been compressed to a pulse duration of 50 fs.
In agreement with the analytical prediction based on the master-equation formalism, this pulse duration has been achieved with the small total resonator dispersion of 0.034 ps 2 . The output pulse spectrum covered a spectral range of 145 nm and corresponded to a Fourier-limited pulse duration of 45 fs. To the best of our knowledge these are the shortest pulses, demonstrated so far, from an ANDF oscillator.

fs pulses from an all-fiber dissipative soliton erbium oscillator
The

All-fiber setup
The all-fiber erbium oscillator is illustrated in In opposite to the all-normal dispersion oscillator demonstrated in the previous chapter, the all-fiber oscillator contains a short section of anomalous dispersion fiber. This anomalous dispersion fiber has to be used because no fiber-based isolator and PBS with normal dispersion fiber pig-tails were available for these experiments. Since the amount of anomalous dispersion is small compared to the normal dispersion (-0.009 ps 2 to 0.033 ps 2 ), its influence on the pulse dynamics is expected to be small.  To further extend all-fiber integration it is possible to replace the prism compressor by an anomalous dispersion fiber section consisting of 25 cm of the SMF-28 fiber. The

Summary
The dissipative solitons [33,41]. In this regard, the dissipative soliton operation and the use of DC gain fibers enabled output pulse energies of up to 31 nJ with average powers of several watts from standard fiber oscillators [26]. Further power scaling was achieved by use of air-clad PCFs with core diameters of up to 70 µm and resulted in output pulse energies close to 1 µJ and pulse peak intensities above 1 MW [27][28][29]63].
However, air-clad PCFs sacrifice the flexibility and simple handling of standard fibers.
They are also difficult to splice to other fibers, which prevents the integration of standard fiber-components such as pump-combiners, spectral filters, and output-couplers into the reported setups. From standard femtosecond fiber oscillators without air-clad PCFs output pulse energies less than 50 nJ have been achieved so far.
The pulse energies of dissipative solitons in mode-locked all-normal dispersion fiber oscillators are limited by the total nonlinear phase-shift accumulated over one resonator round-trip. In order to increase the pulse energy, the peak intensities inside the fiber section of the oscillator must be reduced. This can be done either by using fibers with larger mode-field-diameters, as demonstrated in the PCF-setups [27][28][29]63], or by increasing the pulse duration. However, mode-field-diameters of at most 25 µm are supported by standard step-index large-mode-area (LMA) fibers. On the other hand, the intracavity pulse durations can be increased significantly by integration of long 0.5 µJ pulses from a giant-chirp ytterbium fiber oscillator passive fiber sections into the oscillator. These passive fiber sections increase the total resonator dispersion and result in the generation of so-called giant-chirp pulses with durations of several tens of picoseconds (see [61] and chapter 3). In this chapter, we report on our results obtained by the simultaneous increase of mode-field-diameter and pulse duration in a mode-locked all-normal dispersion step-index fiber oscillator and demonstrate the generation of output pulse energies exceeding 0.5 µJ.

Theory
The increase of the intracavity pulse duration at larger resonator dispersion values can be explained by applying the method of dimensional analysis [64] (see appendix A) to the master-equation (2.12): This master-equation can be solved analytically. However, compared to the more straight forward dimensional analysis, the analytic solutions provide no additional information on the scaling of pulse parameters.
The governing parameters of the master-equation are the difference g − l between the round-trip gain and loss, the spectral filter bandwidth Ω, the resonator dispersion D, the slope of the transmission function of the saturable absorber α, the nonlinearity coefficient integrated over the whole fiber length δ, and the peak power of the intracavity pulses A 2 0 . This set of parameters is responsible, according to dimensional analysis, for the following scaling law for the average intracavity pulse duration τ : where f is a dimensionless function, whose knowledge is not required to explain the scaling of the average intracavity pulse duration. In an all-normal dispersion fiber oscillator setup presented in the next section has been designed to provide the largest total resonator dispersion achievable with the available fiber components, in order to increase the pulse duration and obtain higher pulse energies.

Giant-chirp oscillator setup
The experimental setup of the all-normal dispersion ytterbium fiber oscillator is illustrated in Fig. 6

Results
spectrum by taking the deconvolution factor of 1.48 into account. The pedestal in the autocorrelation function of the compressed pulses contains approximately 13 % of the total pulse energy and results from uncompensated higher-order phase terms.
To verify stable single-pulse operation of the oscillator, the output pulse train has been recorded with a 10 GHz photodiode in combination with a 6 GHz oscilloscope and an rf-spectrum-analyzer. The oscilloscope trace and the radio-frequency spectrum of the photodiode signal are shown in Fig. 6.3 (a) and (b). No satellite pulses are visible in the oscilloscope trace. The radio-frequency spectrum at the fundamental repetition rate of 4.3 MHz has been recorded with a resolution of 1 Hz. The constant heights of the radio-frequency peaks, as well as the noise-suppression of 80 dB, confirm that no Q-switching, period-doubling, or higher-harmonic mode-locking are present. The beam diameter is defined as full width at 1/e 2 of the peak intensity and from the beam caustic a M 2 -value of less than 1.1 can be estimated.
The output pulse energy has been limited by pulse break-up resulting in multiple-0.5 µJ pulses from a giant-chirp ytterbium fiber oscillator pulse operation at higher pump powers. Nonetheless, to the best of our knowledge, the presented results correspond to an increase by a factor of ten in output pulse energy compared to previously reported non-PCF femtosecond fiber oscillators. As predicted by theory, the intracavity pulse durations and the output pulse energies can be further increased using even larger resonator dispersions. However, according to Eq. (4.2) this will also result in compressed pulse durations outside the femtosecond range. Alternatively, the pulse energy can be further increased, without change in pulse duration, by replacing the passive fiber section with normal dispersive fiber Bragg gratings, which reduce the total fiber length and the amount of accumulated nonlinearity. Pulse energy scaling up to the mircojoules level is expected only by use of large-mode area PCFs or distributed spectral filters inside the fiber section as is explained in the next chapter.

Pulse energy limitations in dissipative soliton oscillators
The previous chapters demonstrated the shortest pulse durations and the highest pulse energies per mode-field area, which can be obtained from femtosecond allnormal dispersion fiber oscillators. According to Eq. On the other hand, the pulse energy and the peak power of dissipative solitons increase monotonically with the nonlinear phase shift per round-trip [36]. Additionally, Eq. (4.2) shows that at higher nonlinear phase shifts shorter FTL pulse durations can be obtained. Thus, instead of varying the resonator dispersion, the total nonlinear phase-shift per round-trip can be increased to achieve simultaneously higher pulse energies and shorter pulse durations. Unfortunately, in previously demonstrated setups the nonlinear phase shift has been limited by pulse break-up and onset of multipulse operation. Several publications attributed this limitation to over-driving of the saturable absorber [40,41]. However, no experimental prove for this assumption has been given and no attempts to prevent the over-driving by redesign of the saturable absorber have been reported. Furthermore, no fundamental limitations of the pulse energy and the nonlinear phase-shift per round-trip of dissipative solitons have been identified experimentally or theoretically. Moreover, numerical simulations consistently predict significantly higher pulse energies, and shorter durations than can be obtained experimentally [38,40].
This chapter presents the implementation of polarization maintaining (PM) fiber into a dissipative soliton oscillator mode-locked by NPE. Using the PM fiber section enables to extend the adjustment range of the NPE-based saturable absorber and to prevent Pulse energy limitations in dissipative soliton oscillators the over-driving. As can be seen in Eq. (2.11), the intensity scale of the nonlinear transmission function of the NPE-based saturable absorber is proportional to the length of the non-PM fiber section. Thus, by replacing a part of the non-PM section with PMfiber the over-driving of the NPE can be delayed. In the absence of the over-driving of the saturable absorber, fundamental limitations of the nonlinear phase-shift and the pulse energy will be investigated.

PM-fiber oscillator setup
The setup of the all-normal dispersion ytterbium fiber oscillator with a PM fiber section is shown in Fig. 7

Experimental results
To investigate whether the over-driving of the saturable absorber limits the achievable pulse energies, the length of the SMF section has been varied from 1. Apart from the higher pulse energies, a strong amplitude-jitter of the output pulses, increasing at shorter SMF lengths, can be observed on the oscilloscope. This amplitudejitter in the kHz range appears also in the intensity autocorrelation and usually indicates mode-locked q-switching or multi-pulse operation. However, the oscilloscope Pulse energy limitations in dissipative soliton oscillators  lator setup can be well compared to the oscillator reported in [26].

Discussion
The observed effects can be explained by spontaneous Raman scattering which has already been demonstrated to limit the achievable pulse energies of parabolic pulses in fiber amplifiers [65] but has not been observed in mode-locked fiber oscillators so far. In summary, implementing a PM fiber section into the all-normal dispersion fiber oscillator allows to extend the range of adjustment of the NPE-based saturable absorber.
This is used to prevent the effect of over-driving of the NPE which is identified as a limitation of the pulse energies and nonlinear phase-shifts achievable in dissipative soliton oscillators. However, the absence of the over-driving enables only a small increase of these pulse parameters. The next limitation appears due to a strong jitter of the output pulse train and of the pulse spectrum. This jitter is explained by spontaneous Raman scattering which leads to amplification of a second pulse from the background noise. Thus, to further increase the output pulse energies and reduce the pulse durations of dissipative solitons, the spontaneous Raman scattering must be suppressed by distributed spectral filters inside the fiber section (e.g. long-period fiber gratings).

Free-space fiber oscillators
In chapters 4 and 6, the currently demonstrated pulse duration and energy limitations of femtosecond all-normal dispersion fiber oscillators, which can be realistically integrated into compact and adjustment-free fiber setups, have been presented. In chapter 7 it has been shown that these pulse duration and energy limitations result from overdriving of the NPE-based saturable absorber and the onset of spontaneous Raman scattering.
Recently a new type of mode-locked fiber oscillators has been demonstrated [27][28][29][30], which are best described as free-space fiber (FSF) oscillators. These oscillators combine a free-space resonator with an active fiber section and can be considered as solid-state oscillators in which the laser crystal is replaced by the active fiber. The free-space resonator allows the generation of pulse energies comparable to that of bulk Yb-crystal oscillators, whereas strong gain and nonlinearity in the active fiber section enable dissipative soliton dynamics supporting sub-100 fs pulse durations. Thus, by sacrificing the compactness and adjustment freedom of all-fiber setups, a new range of output pulse parameters from diode-pumped oscillators has been accessed, providing sub-100 fs pulses with energies of several hundreds of nJ [29,30]. In this chapter, we analyze the scaling properties of FSF oscillators and demonstrate pulses with a duration of 31 fs and energies of 84 nJ from an all-normal dispersion fiber oscillator.

Theory
To derive the scaling properties of the all-normal dispersion FSF oscillators, the well- rescaling of the characteristic dimensions). The dimensional analysis has been already Free-space fiber oscillators applied in ultrafast fiber optics to derive the asymptotic parabolic-pulse solutions of normal dispersion fiber amplifiers [44,45]. A major benefit of the dimensional analysis is that an exact mathematical formulation of a physical problem is not necessary required. To obtain useful information or even a complete solution, it is often sufficient to just determine the governing physical parameters. The where L is the active fiber length. The transmission function of the NPE saturable absorber, which is also used as laser output, is given in Eq. (2.11) and determined by three dimensionless parameters α i (i = 1, 2, 3) which are the rotation angles of the three waveplates used for the adjustment of laser polarization [50]. The sinusoidal transmission of the spectral filter is characterized by its full width at half maximum (FWHM) Ω. Thus, the pulse dynamics of this type of all-normal dispersion oscillators depend on a set of eight parameters: β, γ, L, g, Ω, and α i .
By use of dimensional analysis, as described in [64], one can derive the following scaling laws for the electric field envelope A(t,z): where Φ 2 and Φ 3 are also unknown dimensionless functions which can be determined numerically.
According to Eqs.

Experimental setup
In order to investigate the scaling properties predicted above, the experimental setup illustrated in Fig. 8.1 has been used. The fiber section consists only of the ytterbiumdoped double-clad fiber (YDF: Yb1200-10/125DC from Liekki) with a core-diameter of 10 µm, a numerical aperture of 0.08, and an estimated dispersion of 20 ps 2 /km. This fiber has been chosen due to its strong pump light absorption of 6.5 dB/m, which enables the use of fiber lengths as short as 20 cm. The free-space section of the oscillator contains four waveplates in combination with a polarizing beam splitter (PBS) to achieve mode-locking by use of NPE inside the fiber segment. The spectral filter is realized by a birefringent quartz plate (BP) in front of an additional PBS. A free-space isolator ensures uni-directional laser operation and the NPE-rejection port at the first PBS is used as the oscillator output.
Free-space fiber oscillators At short active fiber lengths only a fraction of the total pump light can be absorbed, which results in low laser-efficiencies and output powers. To avoid pump-limitation of the output pulse energy, the repetition rate of the oscillator is reduced using a Herriottcell as optical delay line inside the free-space section. It is important to note that the free-space propagation inside the Herriott-cell does not affect the pulse dynamics inside the fiber section and has no influence on the pulse energy limitations. Furthermore, by use of active fibers with stronger absorptions [30], the Heriott-cell can be removed to achieve the same pulse parameters at higher repetition rates.
Finally, the positively chirped output pulses are compressed using a 300 groove/mm grating compressor with a transmission of 53 %, and the temporal profiles of the compressed pulses have been measured by frequency-resolved optical gating (FROG).
According to the scaling laws (8.1) to (8.3), stepwise reduction of the YDF length and increase of the spectral filter bandwidth have been performed in order to simultaneously increase the output pulse energy and to reduce the compressed pulse durations. The lasing performances of the different realized oscillator setups are listed in

Results
At each YDF length we observed a number of similar operation states with slightly varying pulse parameters. For every oscillator the waveplates were adjusted to achieve  The output pulse spectrum, the intensity autocorrelation of the compressed pulses, and the pulse shape retrieved from the measured FROG trace are shown in Fig. 8.3 for the YDF lengths of 1.95 m, 1.00 m, and 0.32 m. As expected from the dimensional analysis, the optical spectra and temporal pulse profiles of all setups exhibit similar features on different wavelength and time scales. All spectra exhibit the characteristics of dissipative solitons with maxima at the wings of the spectrum and steep spectral side-edges [33,36]. They are asymmetric in shape with a total maximum at long wavelengths and a smaller local maximum at short wavelengths. However, with decreasing fiber lengths more energy is contained in the wings of the pulse spectrum, the energy content of the long wavelength maximum increases, and the center of the pulse spectrum shifts to shorter wavelengths. These effects may be explained by an increasing impact of self-steepening [42], which depends on the spectral bandwidth and has been neglected to allow for the dimensional analysis. All temporal pulse profiles exhibit a main pulse surrounded by a pedestal containing from 12 % up to 53 % of the total pulse energy. These pedestals result from uncompensated higher-order 8.3. Results phase terms accumulated due to nonlinearities in the active fiber. However, we observed no correlation between the energy content inside the pedestal, the deviation from the FTL limited pulse duration, and the YDF length. Therefore, the variation of these parameters can be also attributed to small deviations in oscillator alignments.
In agreement with the dimensional analysis, the shortest pulse durations and the highest output pulse energies have been obtained simultaneously at the shortest YDF length of 0.2 m. At a pump power of 22 W stable single-pulse mode-locking is achieved Free-space fiber oscillators To verify stable single-pulse operation of the oscillator, the output pulse train has been recorded with a 10 GHz photodiode in combination with a 6 GHz oscilloscope and an rf-spectrum-analyzer. The oscilloscope traces and the radio-frequency spectra of the photodiode signal are shown in Fig. 8.4 (e) and (f). No satellite pulses are visible in the oscilloscope trace and in the 150 ps span of the intensity autocorrelator which confirms single-pulse operation.
The radio-frequency spectrum at the fundamental repetition rate of 5.7 MHz has been recorded with a resolution of 1 Hz. The constant heights of the radio-frequency peaks, as well as the noise-suppression of 80 dB, confirm stable mode-locked operation without Q-switching, period-doubling, or higher-harmonic mode-locking [56].

Summary
In summary, scaling properties of mode-locked all-normal dispersion ytterbium fiber oscillators have been studied in terms of pulse duration and energy. We focused on FSF oscillators consisting only of an active fiber and a free-space section without passive fibers. The dimensional analysis has been applied to obtain analytic expressions for the scaling of the laser pulse characteristics. By stepwise variation of the YDF length from 1.95 m to 0.2 m we reduced the compressed pulse duration from 108 fs down to 31 fs and increased the output pulse energy from 31 nJ up to 84 nJ, thereby confirming the theoretical predictions. These parameters correspond to the shortest pulse duration and the highest peak power per mode-field area from an all-normal dispersion fiber oscillator.

Free-space fiber oscillators
The only limitation on pulse duration and energy scaling has been set by the low pump light absorption at short YDF lengths. By use of YDF fibers with stronger cladding absorptions [30] and smaller dispersion values [66] further pulse duration and energy scaling should be feasible. For example, the large-pitch PCF in [30] should allow efficient laser operation at fiber lengths as short as 25 cm. Additionally, this large-pitch PCF has a mode-field diameter of 40 µm, at which the demonstrated peak power of 0.8 MW corresponds to a record high peak power of 13 MW. The highly nonlinear PCF in [66] has a five times smaller normal dispersion at the wavelength of 1 µm than step-index fibers. According to the demonstrated pulse duration scaling, this smaller dispersion values may enable the generation of two-cycle pulses in FSF oscillator setups.

Conclusion
In this thesis, the pulse duration and energy scalings of all-normal dispersion fiber os- Conclusion suggests that peak-power scaling up to several tens of megawatts and the generation of two-cycle pulses may be realized. Therefore, the FSF oscillator concept is expected to significantly expand the operation range of diode-pumped laser systems and to replace more expensive Ti:sapphire lasers in various applications.

Asymptotic parabolic pulse solution of the NLS equation with gain
This section illustrates the method of dimensional analysis [64], which was applied in chapters 6 and 8, using as a well-known example the parabolic pulses. In the following the derivation of the asymptotic parabolic pulse solution to the NLS equation with gain is presented and application of dimensional analysis is explained. The dimensional analysis is used to reduce the number of free variables and to replace the NLS equation by two ordinary differential equations of a single variable. The final parabolic pulse solution is self-similar with respect to propagation distance. Thus, the pulses have constantly a parabolic shape and the pulse parameters scale with the propagation distance inside the active fiber.
The parabolic pulse solution is asymptotic only in the presence of gain (g > 0) and at normal fiber dispersion (β 2 > 0). The NLS equation with gain is given by ∂ 2 A(t, z) ∂t 2 + iγ |A(t, z)| 2 A(t, z), (A.1) and can be split into its real and imaginary parts by the following substitution: A(t, z) = f (t, z) exp(iφ(t, z)), (A.2) where f and φ are both real functions. f is the amplitude of the electric field and φ is its phase. Substituting Eq. (A.2) in Eq. (A.1) results in two equations given by: This set of equations has an infite number of different solutions. Each solution is determined by its initial amplitude and phase distributions f (t, 0) and φ(t, 0).
The first step in dimensional analysis is to identify the important governing parameters. After a certain propagation distance and amount of amplification, any initial pulse will experience strong spectral and temporal broadening and will accumulate a large nonlinear phase-shift. Therefore, it is reasonable to assume that the exact initial amplitude and phase distributions can be neglected in the asymptotic solution. For In the derivation of Eqs. (A.6) the following relation has been used to replace the partial derivatives over z → ∞: where W 0 is the initial normalized pulse energy.
The use of dimensional analysis and neglection of the initial amplitude and phase dis- where α is an arbitrary parameter. From this transformation invariance directly follows the relation: Setting the free parameter α equal to W −1/3 gives: