**2. State-of-the-art**

Laser pulses can be very short if their spectrum is extremely broad. At the high frequency of visible light in the range of Petahertz, it is sufficient to have a spectral width of a few Terahertz to achieve pulse durations of less than one picosecond. This can be realized at very different ways. Very common are quality switched (Q-switched) lasers, which are equipped with a device in the laser resonator that influences – actively or passively – the finesse of the resonator and therefore generates pulsed radiation. In order to achieve even shorter pulses, a number of technological approaches have been realized, making different wavelengths in the laser resonator oscillate synchronously. If the oscillations of the different longitudinal resonator modes, that are supported by the gain medium, have a constant phase relation (i.e. mode locking), a short pulse is created. The more resonator modes couple, the shorter is the pulse.

The needs in optical communications have motivated the development of light sources with short and ultrashort pulse duration significantly. Scientists from atomic, molecular and optical physics made significant contributions and profited from this development. After breaking the picosecond border in pulse duration, an own filed, ultrafast physics evolved. Light pulses of ultrashort duration are a key tool to investigate processes in microcosm. Ahmed Zewail was awarded the Nobel prize 1999 for femtosecond chemistry (Anon 1994). The laws of quantum optics dictate the duration at which processes in atoms, molecules and solids take place (Ferenc Krausz 2009). Electronic transitions in atoms, for example, can happen on a timescale of femtoseconds or attoseconds, where an attosecond (10-18 s)

From photography we know that an image of a moving object can only be unblurred if the exposure time is significantly shorter than the duration of the movement. The timeresolution of the electronic processes mentioned above therefore needs pulse durations of less than one femtosecond. That´s not so easy, since nature does not give us anything for free: Electromagnetic pulses, hence light pulses consist of many waves of different wavelengths which add up in a way that a temporarily confined signal – the pulse – is formed. The envelope within which the electric field oscillates with the carrier frequency defines the duration of the pulse. In order that a pulse can propagate in space, the carrier has to undergo at least one oscillation cycle, which sets a lower limit for the pulse duration. At 750 nm, which is a common wavelength for ultrafast lasers, one oscillation cycle lasts about 2.5 fs. Amplified laser pulses at this central wavelength can be made as short as 3.3 fs, today (A. L. Cavalieri, E. Goulielmakis, et al. 2007). There is not much to be improved in terms of pulse duration at this wavelength. To achieve even shorter pulse durations, shorter wavelengths in the extreme ultraviolet (XUV) or x-ray regime are needed. Unfortunately, nature doesn´t help us here, again. At these wavelengths no materials exist that could be used as laser media, directly. Besides demanding and expensive methods like free-electronlasers (FEL), which are big facilities with hundreds of meters extension, coherent light can be produced by a non-linear wavelength conversion from laser pulses in the visible range. The generation of high-order harmonics (McPherson et al. 1987) (Ferray et al. 1988) provides the possibility to produce attosecond pulses (Paul et al. 2001) (Christov et al. 1997). Before we concentrate on this method, let´s have a look at the formation of ultrashort pulses

compares to one second approximately like a second to the age of the universe.

comprising barely more than one oscillation cycle of the driver wave.

Laser pulses can be very short if their spectrum is extremely broad. At the high frequency of visible light in the range of Petahertz, it is sufficient to have a spectral width of a few Terahertz to achieve pulse durations of less than one picosecond. This can be realized at very different ways. Very common are quality switched (Q-switched) lasers, which are equipped with a device in the laser resonator that influences – actively or passively – the finesse of the resonator and therefore generates pulsed radiation. In order to achieve even shorter pulses, a number of technological approaches have been realized, making different wavelengths in the laser resonator oscillate synchronously. If the oscillations of the different longitudinal resonator modes, that are supported by the gain medium, have a constant phase relation (i.e. mode locking), a short pulse is created. The more resonator modes

**2. State-of-the-art** 

couple, the shorter is the pulse.

After dye lasers had dominated short pulse technology, today nearly all practically all short pulse lasers are based on solids, i.e. glasses and crystals, as the gain medium. In the visible and neighboring infrared range, sapphire doped with titanium ions has prevailed. Ti:Sa lasers have two main advantages – besides being relatively easy to handle: They can be pumped by commercially available, frequency doubled infrared lasers and have a comparably broad amplification bandwidth. To make use of this bandwidth as many modes in the resonator as possible have to be mode-locked and the propagation of a short pulse has to be favored with respect to continuous radiation. Aside from optical components that compensate for dispersion in the resonator, this is mainly made possible by saturable absorbers or concepts based on the optical Kerr effect.

Saturable absorbers are semiconductors with the special characteristic to escalate reflectivity when incoming radiation exceeds a certain threshold intensity. Such a component improves the finesse of a resonator selectively for short pulses and enables mode locking. Another common technology, which can be combined with saturable absorbers, is Kerr-lens modelocking which makes use of the intensity dependence of the refractive index (Kerr-effect) of certain materials. The shorter, i.e. intenser, an incoming pulse is in the material, the stronger self-focusing it experiences. In combination with an aperture such a material ensures that short pulses will be amplified preferably. In combination with chirped mirrors that have been invented in 1990 (Szipocs et al. 1994) it was possible to demonstrate pulse durations shorter than 10 fs with this method. Chirped mirrors (see Figure 2, 3) are coated in a way that different wavelengths penetrated to a differing depth of the layers. The word "chirp" refers to the fact that frequencies - like in the singing of birds – are changing instantaneously. An incoming light pulse whose spectral components are asynchron (chirped) can be compressed to the minimum pulse duration which is supported by its spectrum. Using techniques of this kind, today it´s possible to build oscillators whose optical spectrum comprises nearly a full octave, so that the high frequency part of the spectrum oscillates twice that fast as the low frequency one. In the example discussed here, the spectrum reaches from 450 to 1050 nm, supporting a pulse duration of about 5 fs. The ability to generate ultra-short laser pulses is not the only aspect that puts mode locked lasers in the focus of science. The lasers resonator emits pulses with a repetition rate corresponding to the inverse round trip time in the cavity. This repetition rate therefore depends on the length of the laser cavity and typical values are on the order of 100 MHz. It has to be emphasized that this periodicity in time translates to a periodicity in the Fourier domain: The spectrum of the pulse train emitted by a mode locked oscillator is not continuous but consists of perfectly equidistant spectral lines. This phenomenon was dubbed "frequency comb" and in 2005 Theodor Hänsch was awarded the Nobel Prize for the proposal to use this unique feature for frequency resolved spectroscopy (Reichert 1999) (Telle et al. 1999). Nowadays it is the basis for uncountable applications in the high resolution spectroscopy given that the spacing of the comb lines for a standard mode locked oscillator emitting at 800 nm with a repetition rate of 100 MHz is only 0.00021 nanometers.

Simultaneously the frequency comb found an important application also in the synthesis of ultrashort laser pulses themself. Relying on the measurement of difference frequencies between the individual comb lines on can ensure that every laser pulse emitted has an identical evolution of the electric field underneath the intensity envelope, the so-called "waveform". Figure 1 highlights the importance of this technology for ultrashort laser pulses containing only few optical cycles.

#### Fig. 1. High harmonic generation (HHG)

Ultra intense laser pulses can ionize matter but the released electrons not necessarily leave the atom forever. The "3-step-model" invented by Corkum in 1993 describes this process. If the field strength of the laser pulses is strong enough to suppress the coulomb potential binding the electron to the core, the electrons can tunnel out of the atom (step 1). This free electron is then accelerated in the laser electric field (step 2) that reverts its sign once per period and thus forces the electron to return to its parent ion with a non-vanishing probability and recombine there under the emission of radiation. This process is unlikely but not impossible; the conversion efficiencies reported are about 10-7. Still this technique offers amazing possibilities to the experimentalist due to the unique time structure of the emitted pulses (counted in attoseconds) and the high photon energies that can be achieved (up to keV photons are demonstrated).

Ultra intense laser pulses can ionize matter but the released electrons not necessarily leave the atom forever. The "3-step-model" invented by Corkum in 1993 describes this process. If the field strength of the laser pulses is strong enough to suppress the coulomb potential binding the electron to the core, the electrons can tunnel out of the atom (step 1). This free electron is then accelerated in the laser electric field (step 2) that reverts its sign once per period and thus forces the electron to return to its parent ion with a non-vanishing probability and recombine there under the emission of radiation. This process is unlikely but not impossible; the conversion efficiencies reported are about 10-7. Still this technique offers amazing possibilities to the experimentalist due to the unique time structure of the emitted pulses (counted in attoseconds) and the high photon energies that can be achieved

Fig. 1. High harmonic generation (HHG)

(up to keV photons are demonstrated).

Modern mode locked oscillators achieve Nano-Joule pulse energies which in turn can be focused to intensities in the range of ~1011 W/cm2 insufficient to cause nonlinear response of even ionization of matter (effective ionization of rare gases sets in at laser intensities around 1014 W/cm²).

Pushing ultrafast technology towards intensities above this threshold requires additional efforts (Figure 2). Usually mode locked oscillators are followed by amplification stages that work according to the "chirped-pulse-amplification" scheme (STRICKLAND & MOUROU 1985). The individual spectral components are dispersed and thus the laser pulses are stretched in time leading to drastically reduced peak intensity while maintaining the coherence properties. Subsequently the pulses are amplified in a second laser active medium (crystal or fiber). For this purpose the repetition rate of the oscillator pulse train has to be reduced drastically since the amplifying medium has to fully absorb the thermal energy that goes along with the million fold amplification of the pulses. The difficult handling of the increasing average power and the lack of high-repetitive pump lasers currently render repetition rates higher than a few kilohertz impossible.

For selection of the individual pulses out of the high-repetitive oscillator pulse train contemporary laser systems rely on "pulse-picking" arrangements employing the electro optic effect in a Pockels-cell between orthogonal polarizing filters. After the amplification process (see Fig. 2) the timing of the individual wavelength components is rectified to achieve a short laser pulse by grating- or prism compressors frequently in combination with chirped mirrors.

Unfortunately the amplification process not only reduces the repetition rate but – due to the natural gain window of the laser active medium- also the spectral bandwidth of the laser pulses. To achieve near-single cycle pulses the amplification has to be accompanied by a system that coherently generates additional frequency components on either side of the spectrum. The commonly used systems to achieve that are based on a nonlinear process that was dubbed "self-phase-modulation". The nonlinear interaction of the electric field of energetic laser pulses with rare gases can yield spectra spanning form 450 to 1100 nm covering the whole range of human vision (cp. Fig. 3). By the use of additional chirped mirrors to once again synchronize the individual spectral components state-of-the-art ultrashort pulse laser systems achieve mJ pulse energies within less than 3,5 femtoseconds (A. L. Cavalieri, E. Goulielmakis, et al. 2007) duration. Well focused, no matter can withstand the strength of the electric field that oscillates only 1,5 times during such a pulse reaching intensities up to 1018 W/cm².

If such pulses are used for the generation of high-order harmonics, as mentioned before (see also Fig. 1), one has to distinguish between pulses with many cycles of the driver wave and few-cycle pulses. In the first case, a train of attosecond pulses is generated, which requests that the the sub-cycle electron-light interaction is repeated over several oscillation cycles under exactly the same conditions. Only then, no information is lost during the accumulation of attosecond signals from different cycles of the laser pulse, since the characteristics of the interaction products (e.g. the properties of photoelectrons) are the same. The same request has to be fulfilled when a train of attosecond pulses being separated by half-cycle of the driving laser field is used in a cross correlation with the driving field itself. Even if the laser field had constant amplitude over many cycles, metrology would be restricted to processes being shorter than one half-cycle of the laser field.

Fig. 2. Ultrafast Laser

A continuously pumped mode-locked oscillator emits broadband pulses of fewfemtosecond duration. These pulses are dispersed in time up to ps duration to reduce their peak-intensity and subsequently amplified in a multi-pass amplifier pumped by a multi-mJ Q-switch laser usually operating with kHz repetition rates. Later a prism-compressor rectifies the timing of the different wavelength components reducing the pulse duration again to 10s of femtoseconds. The nonlinear process of self-phase-modulation that takes place in a gas filled hollow-core fiber adds additional spectral components on either side of the amplified spectrum and a set of chirped mirrors converts the octave spanning output spectrum (cp. Figure 3) into ultrashort visible light pulses.

Fig. 2. Ultrafast Laser

A continuously pumped mode-locked oscillator emits broadband pulses of few-

spectrum (cp. Figure 3) into ultrashort visible light pulses.

femtosecond duration. These pulses are dispersed in time up to ps duration to reduce their peak-intensity and subsequently amplified in a multi-pass amplifier pumped by a multi-mJ Q-switch laser usually operating with kHz repetition rates. Later a prism-compressor rectifies the timing of the different wavelength components reducing the pulse duration again to 10s of femtoseconds. The nonlinear process of self-phase-modulation that takes place in a gas filled hollow-core fiber adds additional spectral components on either side of the amplified spectrum and a set of chirped mirrors converts the octave spanning output

#### Fig. 3. Ultrashort Laser pulses

As a consequence of the uncertainty relation short laser pulses require an extended spectrum. To synthesize laser pulses consisting of little more than one cycle of the electric field the spectrum has to span an octave (it has to coherently contain frequencies between υ and 2υ ). Given the human eyes sensitivity being restricted to a range between 400 nm and 750 nm, single cycle laser pulses appear white as seen on the photograph that shows the scattered light if such pulses are reflected off the multilayer mirrors mentioned in the text. The graph compares the measured spectrum of such pulses with the sensitivity range of human sight (rainbow shaded area). The plot below highlights how for such short pulses the phase (carrier-envelope-phase CEP) between the intensity envelope function and the oscillations of the carrier wave defines the waveform and thus the action of the pulse in nonlinear processes.

Multi-cycle attosecond metrology (Ferenc Krausz 2009) can be used to investigate periodically-repeated recollisions which are driven by the multi-cycle light wave and trigger the process under study in a correlated manner in the same interaction within each wave cycle. One of the two correlated processes can serve as a clock for the other. Inventing this concept, (Niikura et al. 2002) demonstrated that the vibrational motion of a molecule triggered by ionization clocks the recollision electron and (Niikura et al. 2003) showed the case vice versa for a diatomic molecule.

In most laser systems, multi-cycle pulses have a Gaussian envelope which results in attosecond pulses with varying intensity and photon energy within the pulse train generated, If these pulses trigger even simple processes a retrieval of the process gets highly complicated.

Therefore, employing few-cycle pulses for the synthesis of high-order-harmonics (Fig. 1) becomes highly significant. Given a smart choice of the phase between the intensity envelope and the oscillations of the carrier wave, waveforms can be generated where the intensity contrast between adjacent maxima of the electric field is maximized.

Since every single half cycle contributes to the emission of high-harmonics the highest intense half cycle apparently leads to the emission of the highest energetic photons. Filtering only this radiation out of the emitted spectrum thus yields coherent X-ray pulses with a duration corresponding to only a fraction of the driving lasers half-cycle duration. The shortest light bursts demonstrated ever have been generated in this manner: Lasting only about 80 attoseconds they contain roughly 108 photons per pulse at a central wavelength of 12 nm deep in the ultraviolet.

As has been discussed, the parameters of the driving laser pulse are extremely important for the characteristics of the generated harmonic radiation. Pulse duration, pulse energy, photon energy, and the number of attosecond bursts (single pulses or a train) are given by the properties of the driving laser pulse. In terms of the photon energy, the wavelength of the driving pulse is crucial. It may be counterintuitive on first glance but longer wavelength driver pulses are able to generate harmonics at higher photon energy (Vladislav S. Yakovlev et al. 2007) since the electron can accumulate more energy during its trip to the continuum (in the semi classical picture). Therefore the development of few-cycle laser pulses in the infrared spectral range is pursued by several groups in the world.

(Fuji et al. 2006) have demonstrated an OPA system whose pulses comprise only a few oscillation cycles at 2.1 μm carrier wavelength. IR-driven HHG was pioneered by L'Huillier and co-workers (L'Huillier & Ph. Balcou 1993) and DiMauro and co-workers (Sheehy et al. 1999) followed by studies of (Bellini 2000), (Shan & Chang 2001). As pointed out by DiMauro, the favourable scaling of the maximum photon energy with driver wavelength may open the way to generating coherent light and attosecond pulses at photon energies substantially beyond the kiloelectronvolt frontier, which was reached recently with NIR few-cycle light (J. Seres et al. 2005). This follows from the fact that the cutoff, i.e. the highest achievable energy in HHG (see Fig 1) scales with the ponderomotive potential - the cycle averaged quiver energy an electron gains in an oscillating field - and therefore with the square of the driving wavelength.

Another technique that aims at increasing the flux of HHG especially at high photon energies is the so-called quasi phase-matching (QPM). In this approach one tries to overcome a problem that limits the conversion efficiency between driver and harmonic radiation. Dispersion in the non-linear medium used for HHG results in a phase mismatch between the driver wave in the visible and NIR spectral range and the short wavelength product of the conversion process. This leads to a destructive interference after a certain distance and reduces the flux of the harmonic radiation. Methods like using a modulated fiber (Christov et al. 1998) as a cell for the conversion process or using several targets to modulate the density of the conversion medium aim at bringing the both waves involved in cadence again. Especially for photon energies in the keV range, where the phase mismatch plays a crucial role, this method is of great importance.

Therefore, employing few-cycle pulses for the synthesis of high-order-harmonics (Fig. 1) becomes highly significant. Given a smart choice of the phase between the intensity envelope and the oscillations of the carrier wave, waveforms can be generated where the

Since every single half cycle contributes to the emission of high-harmonics the highest intense half cycle apparently leads to the emission of the highest energetic photons. Filtering only this radiation out of the emitted spectrum thus yields coherent X-ray pulses with a duration corresponding to only a fraction of the driving lasers half-cycle duration. The shortest light bursts demonstrated ever have been generated in this manner: Lasting only about 80 attoseconds they contain roughly 108 photons per pulse at a central wavelength of

As has been discussed, the parameters of the driving laser pulse are extremely important for the characteristics of the generated harmonic radiation. Pulse duration, pulse energy, photon energy, and the number of attosecond bursts (single pulses or a train) are given by the properties of the driving laser pulse. In terms of the photon energy, the wavelength of the driving pulse is crucial. It may be counterintuitive on first glance but longer wavelength driver pulses are able to generate harmonics at higher photon energy (Vladislav S. Yakovlev et al. 2007) since the electron can accumulate more energy during its trip to the continuum (in the semi classical picture). Therefore the development of few-cycle laser pulses in the

(Fuji et al. 2006) have demonstrated an OPA system whose pulses comprise only a few oscillation cycles at 2.1 μm carrier wavelength. IR-driven HHG was pioneered by L'Huillier and co-workers (L'Huillier & Ph. Balcou 1993) and DiMauro and co-workers (Sheehy et al. 1999) followed by studies of (Bellini 2000), (Shan & Chang 2001). As pointed out by DiMauro, the favourable scaling of the maximum photon energy with driver wavelength may open the way to generating coherent light and attosecond pulses at photon energies substantially beyond the kiloelectronvolt frontier, which was reached recently with NIR few-cycle light (J. Seres et al. 2005). This follows from the fact that the cutoff, i.e. the highest achievable energy in HHG (see Fig 1) scales with the ponderomotive potential - the cycle averaged quiver energy an electron gains in an oscillating field - and therefore with the

Another technique that aims at increasing the flux of HHG especially at high photon energies is the so-called quasi phase-matching (QPM). In this approach one tries to overcome a problem that limits the conversion efficiency between driver and harmonic radiation. Dispersion in the non-linear medium used for HHG results in a phase mismatch between the driver wave in the visible and NIR spectral range and the short wavelength product of the conversion process. This leads to a destructive interference after a certain distance and reduces the flux of the harmonic radiation. Methods like using a modulated fiber (Christov et al. 1998) as a cell for the conversion process or using several targets to modulate the density of the conversion medium aim at bringing the both waves involved in cadence again. Especially for photon energies in the keV range, where the phase mismatch plays a crucial role, this method is of great

intensity contrast between adjacent maxima of the electric field is maximized.

infrared spectral range is pursued by several groups in the world.

12 nm deep in the ultraviolet.

square of the driving wavelength.

importance.
