Areviewofultrafast optics and optoelectronics

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1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS J. Opt. A: Pure Appl. Opt. 5 (2003) R1 R15 PII: S (03) REVIEW ARTICLE Areviewofultrafast optics and optoelectronics Günter Steinmeyer 1 Institut für Quantenelektronik, ETH Zürich Hönggerberg, CH-8093 Zürich, Switzerland steinmey@mbi-berlin.de Received 21 March 2002, in final form 30 July 2002 Published 8 November 2002 Online at stacks.iop.org/jopta/5/r1 Abstract The speed of optoelectronic devices is normally limited by the components used on the electronic side of the device. The direct generation of short light pulses from short current pulses, for example, is limited by the speed of an electronic pulse generator or the response time of a laser diode. These electronic bandwidth limitations can be overcome by switching to indirect schemes. These schemes use optical means, whenever bandwidth is an issue. This is combined with much slower optoelectronic technology, bringing together the inherent speed of all-optical approaches and the virtues of standard optoelectronics. Apart from the generation of short pulses, we will also address their detection and characterization, their modulation, and transmission effects. These methods carry the functionality of optoelectronics from a temporal resolution of a few picoseconds well into the femtosecond range. Keywords: Femtosecond pulses, ultrafast lasers, pulse characterization, chirped mirrors, dispersion, ultrafast optoelectronics, modulators (Some figures in this article are in colour only in the electronic version) 1. Introduction Optoelectronics is based on electronic devices that are used for emitting, modulating, transmitting or sensing light. At avery fundamental stage, these devices require interaction of light with an electronic current, converting photons into electrons or vice versa. The temporal resolution of an optoelectronic emitter, for example, is therefore always limited by the fastest available rise time of a currentpulse generator. Similarly, detection of light in a photodetector can only be accomplished directly with a temporal resolution of a few picoseconds [1]. On the electronics side, additional constraints can be imposed by parasitic inductances or capacitances in the electronic circuitry and high-frequency damping mechanisms in microwave cables. Streak cameras [2], merging the generation of photoelectrons 1 Presentaddress: Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Str. 2a, D Berlin, Germany. and their temporal resolution into one device, overcome some limitations. Nevertheless even these fastest direct optoelectronic detection devices are typically limited to a response time of the order of 1 ps. The examples discussed so far rely on adirect interaction of light and an electronic current. In the following we will describe ways to circumvent the electronic bandwidth problem. The fundamental idea behind ultrafast optoelectronics becomes clear from the development of optical communication networks. In early fibre optic data links, light was converted back into an electronic current prior to any processing. This concept is being increasingly replaced by all-optical means ofprocessing photonic data streams. A major improvement in terms of data capacity has been achieved by the method of wavelength-division multiplexing, which allows for terabit/s rates by transmitting many channels at different wavelengths through one and the same fibre. In this section, we will introduce methods to provide the fundamental optoelectronic functionality of /03/ $ IOP Publishing Ltd Printed in the UK R1

2 emitting, modulating, transmitting and sensing light with a temporal resolution of a few femtoseconds. These methods are ultimately limited only by the duration of the optical cycle itself. We will refer to these schemes as ultrafast optoelectronic devices, even though the individual optoelectronic components, mediating between photons and electrons, can be inherently slow. The methods described split the optoelectronic process into two steps, an ultrafast alloptical step, which ensures sufficient bandwidth, and a second slow electronic step to allow for efficient conversion between photons and electrons at a strongly reduced bandwidth. This review is organized as follows: first we review methods for the generation of femtosecond pulses. These sources are actually driven by a continuous light or current source and shape a femtosecond pulse employing only optical nonlinearities. Femtosecond pulses can cover a vast bandwidth of several hundred terahertz. Short light pulses also experience reshaping effects, which may modify their pulse shape. Compared with electronic pulses, however, these effects only set in on terahertz rather than gigahertz bandwidth scales, and they also mainly affect the spectral phase rather than the amplitude. Therefore, particular attention will be paid to methods that allow compensation of dispersive pulse broadening. This discussion will be followed by an overview of femtosecond pulse characterization methods. In the fourth section, finally, we will address methods to modulate the phase and amplitude of femtosecond pulses. Together with the characterization methods, ultrafast optoelectronics nowadays allows for synthesis of desired pulse shapes and control of optical waveforms, very similar tothegeneration of arbitrary electronic waveforms. 2. Ultrafast laser pulse generation Unlike in electronics, bandwidth is typically abundant in optical systems. This is a clear driving force behind all-optical telecommunications. Fibre-optic systems can provide terabit/s of transmission capacitance over transatlantic distances [3, 4]. From elementary Fourier theorems, it is clear that the widest bandwidth also supports the shortest pulse. The laser material with the widest known bandwidth is titanium-doped sapphire [5]. The nm gain bandwidth directly supports a pulse duration of less than 5 fs, corresponding to less than two optical cycles of the electric field. Laser operation is generally sustained by an optical cavity around the gain material. This cavity provides the optical feedback. The photons circulating in the cavity experience laser gain and losses due to output coupling [6]. Laser gain saturation favours equal filling of the cavity with photons, i.e. continuous wave (cw) operation of the laser [7]. In order to generate a short pulse, the energy content of the optical cavity has to be temporally confined into as short an interval as possible. This requires us to introduce a mechanism that favours short-pulse operation of the laser. One means of doing so is the insertion of an intracavity amplitude modulator, which opens and closes in synchronicity with the light travelling through the cavity [8] (compare figure 1). This effectively reduces the losses of those photons travelling synchronously with the fully open position of the modulator. If conventional electro-optic or acousto-optic modulators are Figure 1. Schematic illustration of continuous wave (cw, top) and mode-locked (bottom) operation of a laser. In the cw case, the cavity is equally filled with photons, i.e. the energy density E is constant. Insertion of a modulator synchronously driven at the cavity round-trip frequency focuses light into a small time slot centred at the fully open position of the modulator (bottom). The two situations are also depicted in the frequency domain on the right-hand side. Continuous operation results in single-longitudinal mode operation of the laser. The modulator creates sidebands at the neighbouring modes, effectively transferring energy to the spectral wings. This is called mode-locking. used, this whole scheme is always limited by the electronic pulses of the modulator driver, even though pulses shorter than thedriving pulses can be generated. Following the philosophy of the introduction, it is therefore necessary to eliminate the bandwidth limitation by switching to an all-optical modulator Saturable absorber mode-locking The simplest all-optical modulator schemes rely directly on saturable absorption of an optical material [9 11]. Typically, either dyes or semiconductor materials are used. At high intensities the absorption of these materials bleaches out, because many carriers have been excited from the ground statetohigher energy states. Such an all-optical modulator is automatically synchronous with the light in the cavity. A small initial power fluctuation inside the cavity will experience less loss than the rest of the cavity s energy content and will therefore self-amplify until all the energy is concentrated in a small time slot. Unfortunately, the relaxation of the bleaching is not arbitrarily fast. Again, pulses shorter than the modulation time constants can be produced, and it is not even necessary that the relaxation time of the absorber is faster than the cavity round trip. Nevertheless, the all-optical method also experiences a limitation from bandwidth effects. Typically, this forbids the generation of pulses much shorter than apicosecond, even though some clever schemes have been worked out that overcome some of the limitations of a slow absorber and extend operation of a saturable absorber mode-locking well into the femtosecond range [12 14]. Some of these approaches are fairly specialized and rely on the interplay of several mechanisms inside the cavity. An alternative approach came from the use of so-called reactive nonlinearities, e.g. the Kerr nonlinearity (see figure 2). Other than saturable absorption, reactive effects influence the phase of the light only, but not its intensity profile. This self-phase modulation causes a delay of high intensities with respect ω ω R2

3 Figure 2. The nonlinearopticalkerreffectiscausedby a dependence of the index of refraction on intensity. Along the axis of propagation z,thiscauses a phase retardation of the most intense part of the temporal pulse profile. This effect is also called self-phase modulation. In the plane perpendicular to z,theretardation causes a deformation of the phase fronts. In the central part of the spatial beam profile the phase front experiences an additional curvature, i.e. the Kerr effect causes an effect similar to a lens. Therefore the transverse Kerr effect is also referred to as a Kerr lens. to low intensities [15]. As no carrier dynamics is involved, this mechanism can be very fast with a response time of less than 1 fs. The reactive character of the nonlinearity calls for a conversion mechanism transferring the nonlinear phase modulation into an effective amplitude modulation Passive mode-locking based on reactive nonlinearities Use of reactive nonlinearities leads us to the concept of the effective saturable absorber. Such a device is actually a combination of a phase-to-amplitude converter and a reactive nonlinearity. Several effective saturable absorbers have been proposed. One method is to place the self-phase modulator in asecond cavity coupled to the gain cavity as the converter. The coupled cavity acts as a nonlinear mirror providing high reflectivity for the high-intensity centre part of the pulse and low reflectivity everywhere else. This method is called additive-pulse mode-locking [16]. Another method is Kerrlens mode-locking [17 19]. Here the transverse effect of the Kerr nonlinearity is used instead (see figure 2). When the central and most intense spatial portion of an optical beam is phase-retarded, this corresponds to the focusing action of a lens. A suitable arrangement of intracavity apertures can then be used to translate the Kerr nonlinearity into an effective absorber. The focused high-intensity light experiences fewer losses at these apertures. Alternatively, one can also arrange the cavity for a better overlap with the pump light when the additional nonlinear lens is in place. Pulses as short as approximately 5 fs have been produced with this method [20, 21] Amplification Typical KLM lasersdeliverpulse energies of afew nanojoules at a repetition rate of 100 MHz. Pulse energies can be increased either by using a longer cavity [22], by additional extracavity amplification [23] or by cavity dumping [24]. All these methods allow for a pulse energy of a few 10 nj at MHz repetition rates. Further increase of the pulse energy into the µj range while still maintaining femtosecond time duration would increase pulse peak powers into a regime where most optical materials would encounter damage. For further amplification, the pulse therefore needs to be stretched before amplification and is then recompressed to its original duration after the amplifier. The stretcher and compressor will be treated in the following section. The method is called chirped pulse amplification (CPA) [25, 26] and has been demonstrated with 15 fs pulses of millijoule energy at 10 khz repetition rate [27, 28]. Even stronger amplification and reduction of repetition rate can lead to pulse energies of hundreds of joules at 450 fs pulse duration [29]. The latter system reaches a peak power of 1.5 PW. The focused intensity reaches Wcm 2.Thefieldstrength in the focus of such a laser exceeds typical inner atomic binding forces by about three orders of magnitude [30] Wavelength conversion from THz to x-rays So far we have concentrated on direct oscillator and amplifier schemes, relying on a laser transition and its gain. There exist a wide variety of laser materials that can be used for Kerr-lens mode-locking, other schemes of passive mode locking and also for amplification [13, 31 33]. In certain spectral regions, e.g. theblue or UV, it is virtually impossible to find materials with similar properties to, for example, Ti:sapphire, which is probably the most successful material for oscillators and amplifiers. An alternative approach is to convert the wavelength range of the femtosecond radiation by nonlinear optical mechanisms rather than implementing lasers for different wavelength regimes. For the conversion process, nonlinear optical mechanisms are used. One possibility is simple frequency-doubling, which has been demonstrated down to about 5 fs duration [34, 35]. Another method is optical parametric amplification, which can be used both in the near-infrared or visible spectral range [36, 37]. Other methods leading deeper into the UV are Raman side-band generation [38] and high-harmonic generation [30, 39, 40]. The latter method holds the potential to even generate pulses with attosecond time signatures in the XUV range [41 43]. High-harmonic generation has been demonstrated down to a wavelength of a few ångstroms [44]. Even shorter wavelengths can be produced by Thomson scattering [45]. Femtosecond x-ray pulses produced by the latter method allow for structural studies of laser-induced melting at extremely short time scales. Wavelength conversion is also a very important mechanism to generate radiation of much longer wavelength than Ti:sapphire lasers. The near-infrared spectral range can be addressed by difference frequency generation [46] and parametric conversion [47], very similar to the abovedescribed methods of visible femtosecond generation. In the most extreme case, called terahertz radiation [48], difference frequency generation takes place between different spectral components of the mode-locked laser pulse itself. For the conversion process, either bulk crystals or specially designed transducer structures are used. The generated electromagnetic waveform typically covers Fourier components from gigahertz to several terahertz [49]. Pulses consisting of less than one R3

4 optical cycle can be generated. The energy of the individual terahertz pulses is typically very low. Photoconductive sampling allows for very sensitive measurement of the THz waveform by gating the waveform with a second laser pulse. This method is called THz time-domain spectroscopy [48]. Similarly tothediscovery of x-rays, THz radiation generated by femtosecond lasers opened a new spectral window, which could not easily be accessed by any other means. Therefore, THz radiation is also called T-rays [50]. 3. Femtosecond pulse propagation effects and dispersion compensation In microwave electronic systems, a severe limitation is imposed by high-frequency damping mechanisms. In optics, an absorptive bandwidth limitation is typically not a concern or can be easily avoided. Many dielectric media, such as glasses and crystals, are transparent in the range from 150 to 1000 THz [51]. Limitations typically only arise in optical amplification or nonlinear optical conversion. In a 10 THz window in the near infrared (1.55 µm wavelength), exceptionally low losses of 0.3 db km 1 have been demonstrated in silica fibres [52]. This ultralow-loss window has received particular attention as it coincides with the amplification bandwidth of Er-doped glass amplifiers [53, 54], which can be easily embedded into optical telecommunication systems. Compared with electronic systems, therefore, optical bandwidth is abundant and a much lesser concern Group delay dispersion as a leading-order propagation effect If one tries to launch a 100 fs pulse train into a fibre, however, one would find that the pulses already broaden to picoseconds after a few metres of propagation because of dispersion. Dispersion causes different spectral components of the pulse to propagate at different group velocity, which induces broadening of pulses during propagation. Compensation of dispersive effects therefore poses a ubiquitous problem not only in telecommunication systems [3], but also in ultrashort pulse generation [55]. A pulse with angular carrier frequency ω = 2πc/λ experiences a group delay (GD) GD(ω) = l d ω n(ω) (1) dω c when propagating through a dispersive medium with index n and length λ. TheGDdetermines the propagation time of a pulse and must not be confused with the phase delay ln/c. To leading order, pulse broadening is governed by the group delay dispersion (GDD) GDD(ω) = l d2 ω n(ω). (2) dω 2 c Gaussian pulses ofduration τ 0 are stretched to duration ( ) GDD 2 τ = τ 0 1+ (3) when propagating through a material with dispersion GDD. Equation (3) is a useful relation to estimate the severity of τ 2 0 GDD (fs 2 /mm) refractive index GD (ps/mm) UV resonance anomalous dispersion normal dispersion positive dispersion wavelength (nm) IR resonance negative Figure 3. Material dispersion. A schematic diagramto show the dispersion of a dielectric material (simplified model of fused silica) with a UV resonance at 80 nm and a vibronic IR resonance at 10 µm. The top figure shows the refractive index itself. For the entire range between the resonances, dn/dλ <0 holds, which is referred to as normal dispersion. At resonance, we find anomalous dispersion dn/dλ >0. Also shown are the resulting GD and GDD per millimetre material path. The GDD is the leading term responsible for pulse reshaping during propagation. Note that regions of positive and negative GDD do not coincide with those of normal and anomalous dispersion. pulse broadening in optical systems [3]. Optical materials can exhibit both negative or positive dispersion (see figure 3). Fused silica, for example, shows a positive dispersion below approximately 1.3 µm andnegative dispersion above. This general behaviour is illustrated in figure 3. Note that GD and GDD are shown per millimetre material path. At the zero dispersion wavelength of a material, broadening effects due to GDD are eliminated; but similar effects are then caused by higher-order derivatives of the refractive index, e.g. third-order dispersion (TOD) TOD(ω) = l d3 ω n(ω). (4) dω 3 c Positive and negative GDD must not be confused with normal or anomalous dispersion 2,whichusually refers to the sign of the first derivative dn/dλ (see figure 3). In a system with positive dispersion, blue spectral components will be retarded relative to red components. This causes a dependence of the pulse carrier frequency versus time, which is usually referred to as a chirp. Depending on the dispersion that the 2 Some authors also use the words normal or anomalous when referring to group delay dispersion. As this has given rise to much confusion, we will strictly use the words positive or negative when referring to the GDD. R4

5 pulse has experienced, one talks about positive or negative chirp. If no other mechanism is present, both signs of dispersion are totally equivalent in terms of pulse broadening. However, self-phase modulation will typically generate a positive chirp. As self-phase modulation also introduces spectral broadening, its combination with negative dispersion allows for pulse compression schemes. Balancing a positive chirp generated by the nonlinear optical process of self-phase modulation and negative material dispersion can also lead to self-stabilizing optical pulses called solitons. These solitons can propagate over great distances through dispersive systems without changing their pulse shape [3, 56]. The discussion so far makes it clear that the control and engineering of dispersion is of paramount importance in ultrafast optical systems and telecommunications. Dispersion management is important for long-distance fibre links [57, 58]. With ever-wider bandwidth becoming accessible, compensation of higher-order dispersion becomes a consideration [59]. Pulse compression is a major mechanism in ultrashort pulse generation [15, 60]. Passively mode-locked lasers make extensive use of recompression of pulses, employing self-phase modulation in the laser crystal together with negative dispersion in the cavity [61, 62]. Only if this mechanism is fully exploited can the shortest pulses be generated. In general, an ultrafast pulse compressor can only be built with negative dispersion, which unfortunately is not available from optical materials at below 1 µmwavelength. This calls for alternative concepts to compensate for material dispersion and chirps caused by nonlinear optical mechanisms. One can classify sources of dispersion into bulk dispersion (i.e. from homogeneous materials like glasses and crystals), geometrical dispersion (prism and grating arrangements), dispersion from interferometric effects and microstructured dispersion (fibre Bragg gratings, chirped mirrors, chirped quasi-phase-matched crystals, arrayed-waveguide gratings (AWG)). Bulk dispersion has already been treated above. Reference data for many materials can be found in [51] Geometric dispersion prism and grating compressors In the following we will first address geometrical dispersion as produced by prism [63, 64] and grating sequences [65]. When ashort pulse is sent into a prism, e.g. its spectral components are angularly dispersed and sent into different directions (see figure 4). A second prism of opposite alignment can then be used to make the spectrally dispersed beams parallel again. On their propagation between both prisms, the outer rays have experienced adelay relative to those at the centre. It is important tonote that this parabolic spectral delay is equivalent to negative GDD. It can therefore be used to compensate for positive material dispersion. Pairs of Brewster-cut prisms can compensate dispersion without introducing losses and have been very successfully used inside laser cavities [63]. The major shortcoming of the geometrical approach, however, is that it introduces higher-order dispersion terms. For prism compressors, a careful choice of the prism material allows for vanishing TOD in the wavelength range above 800 nm [66]. In particular, fused silica prism pairs introduce vanishing third-order aberrations in the Ti:sapphire wavelength range, which was used for the first demonstration of sub-10 fs pulse generation with this laser [67]. Combinations of four and more prisms, such as the Proctor Wise prism sequence [68], have been investigated. To some extent, the latter approach allows for a control of the ratio of second-order to third-order dispersion. Grating sequences may be used instead for the same purpose (see figure 4). They are of extreme importance for CPA [25, 69], which allows for the amplification of pulses from the oscillator to the millijoule or joule level at a reduced repetition rate. To prevent damage in the amplifier chains, the oscillator pulse is stretched into the picosecond range before amplification. This reduces its peak power by the stretching ratio and also prevents nonlinear optical effects. After amplification the pulse can then be recompressed into the femtosecond range using a grating sequence with exactly opposite dispersion of the stretcher and the amplifier material dispersion. This restores a short pulse duration and allows for the generation of extremely high peak powers. The stretcher [63] employed in CPA is typically a grating sequence, which incorporates atelescope with 1magnification. The telescope exactly inverts the dispersion of acompressor in all orders. The trick is now to slightly unbalance a stretcher and a matched compressor and to accommodate material dispersion in the difference between stretcher and compressor dispersion. Second-order dispersion in equation (2) is adjusted by a difference in grating distances, third-order dispersion in equation (4) can be zeroed out by adjusting grating angles [70]. Fourth-order aberrations, finally, can be compensated for by use of gratings with different groove frequencies [71]. Typically, aberrations of the telescope have to be compensated for by suitably corrected optics, see e.g. [72]. This approach has been used for the demonstration of a 15-fs amplified pulse duration [27, 28]. Other approaches exist, which introduce acontrolled amount of imaging aberrations in the stretcher s telescope to achieve compensation up to fourth order [73] Microstructured dispersion AWGs and chirped mirrors One of the major shortcomings of the geometrical dispersion compensation approaches is that they typically only allow for compensation of second-order dispersion with the few exceptions already noted. Therefore geometrical dispersion compensation schemes are limited to approximately 100 THz bandwidth. However, the idea of the prism compressor can be readily extended to compensation of arbitrary dispersion. Rather than using free-space propagation of laser beams, one could imagine coupling each and every spectral component into an individual fibre with well-engineered length. A discrete approach would certainly be cumbersome, however integrated optical devices similar in function havebeen demonstrated and arereferred to as AWG [74 76]. This type of device is the most pictorial example for microstructured dispersion compensation and is shown in figure 5. Rather than directly introducing a wavelength-dependent propagation length, several other methods for arbitrary dispersion compensation are also shown in figure 5. One of these approaches is chirped mirrors [77 79]. These dielectric mirrors consist of alternating pairs of transparent high-index and low-index layers. The same effect can be achieved in optical fibres by modifying the refractive index with exposure R5

6 λ Figure 4. Geometric dispersion as caused by prism and grating sequences. Both prisms and gratings exhibit angular dispersion. A beam with a broad input spectrum is dispersed into different directions of propagation, with all beams originating at one and the same location at the tip of the first prism. Positions with equal phase delay therefore describe a circle centred at the prism tip. To leading order, the phase delay relative to the centre beam is parabolic. The second prism only serves to render all beams parallel again. Typically, such an arrangement is used in double pass with a retroreflecting mirror as shown. The beam propagating exactly from tip to tip marks the short-wavelength horizon of the prism compressor. A parabolic phase delay front corresponds to a linear GD (as shown in the grating compressor). An exact calculation for the group and phase delay of a prism compressor (fused silica Brewster prisms with 1 m apex distance) based on [64] is also shown. The resulting GDD is depicted below. Note the strong higher-order dispersion of this approach. Figure 5. Microstructured dispersion compensation. Shown are four different concepts that can compensate for arbitrarily shaped dispersion. Top: arrayed waveguides (AWG). The coupling sections provide wavelength-dependentcoupling into waveguidesof different length. This makes the path length a designable function of wavelength. Second: chirped mirrors. Here the Bragg wavelength is varied over the mirror stack, making the penetration depth a function of wavelength. Impedance matching sections are required to reduce detrimental interferometric effects. Third: fibre Bragg gratings. They work similarly to chirped mirrors, but achieve impedance matching by a different apodization method, as shown on the right. Bottom: Chirped grating quasi-phase matching. Here the input light is converted into the second harmonic in a quasi-phase-matched crystal. The poling period of the crystal determines the conversion wavelength, which again allows the total GD to be controlled as a function of wavelength. to UV light through a periodic mask [80, 81]. Those portions of the fibre that have been exposed to the short-wavelength radiation show a modified index of refraction. Even though the index differences are much smaller than in the chirped mirror approach, fibre Bragg gratings provide the same functionality as a distributed Bragg reflector if the period of the index modulation is chirped along the fibre [59, 82, 83]. A Bragg mirror reflects light when all Fresnel reflections at the high/low index interfaces constructively add up. This is the case when the optical thickness of all layers is chosen equal to a quarter of the light wavelength. Varying the optical layer thickness along the mirror structure then results in a dependence of Bragg wavelength λ B on penetration depth z. Chirping the mirror structure therefore allows the generation of a wanted group R6

7 delay, GD(ω). 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. It needs to be mentioned that this simple picture is often distorted by other contributions to the dispersion. In chirped mirrors, the top reflection at the interface to air gives rise to undesired interferences which spoil the dispersion characteristics of the mirror and cause strong spectral fluctuations of the dispersion. A solution to this problem is impedance matching from the ambient medium to the mirror stack. In deposited mirror structures, a partial solution can be provided by double-chirping [78, 79]. Other methods have been proposed to overcome these dispersion fluctuations [84, 85]. In fibre optics, UV exposure can be slowly reduced for a slow increase of the index modulation in the initial part of the fibre Bragg grating. This is referred to as apodization [86]. Anovel approach to arbitrary dispersion compensation is also offered by quasi-phase-matching (see figure 5). Here the conversion point in a nonlinear conversion process is changed with propagation distance. This offers a means to tailor processes such as second-harmonic generation to support extremely broad bandwidths [35, 87] Interferometric effects Gires Tournois interferometers The dispersion of chirped mirrors stems from the interference of many reflections at the interfaces between high- and lowindex materials. An alternative approach to interferometric dispersion compensation is the use of Gires Tournois interferometers (GTI) [60]. Suchaninterferometer consists of a partial reflector and a high reflector. This combination reflects all incoming light and has a spectrally flat amplitude response. Its phase response, however, exhibits resonances spaced by ν = c/2l, similar to a Fabry Pérot interferometer. The spectral phase is a periodic function with regions of negative and positive dispersion. The GTI can be built from discrete components, but can also be implemented as a mirror structure with a relatively thick spacer layer between quarter-wave sections for the partial and high reflector. These structures have been successfully used in femtosecond oscillators [88 90]. Compared to chirped mirrors, they typically exhibit a lower bandwidth, but they can provide larger values of negative GDD. Their manufacture is not quite as demanding as that of chirped mirrors, and GTI mirrors can also reach very high values of reflectivity [90]. Therefore this concept is interesting for lasers with much intracavity dispersion working at longer pulse durations than cavities with chirped mirrors. 4. Measurement of optical waveforms with femtosecond resolution 4.1. Autocorrelation The major problem in characterizing optical waveforms lies in the fact that optical pulses are among the shortest manmade events and there simply exists no shorter controllable event to sample the waveform. Quite naturally, therefore, all earlycharacterizationmethods employed autocorrelation [91], which characterizes a laser pulse using one and the same pulse as the sample pulse and the pulse to be sampled. Using one replica of the input pulse as the reference sample, a second replica is delayed relative to the reference and then multiplied with the reference pulse (see figure 6). Technically, the multiplication of the two optical signals is performed using a nonlinear optical effect such as second-harmonic generation or two-photon absorption [92]. As a result, the autocorrelation trace AC(δt) = I (t)i (t δt) dt (5) is measured, where I (t) = E(t)E (t) is the optical intensity [93, 94]. This type of autocorrelation is called background-free, as it will measure zero signal for large delays δt ±. Abackground-free autocorrelator uses a noncollinear beam geometry in such a way that secondharmonic generation (SHG) requires one photon from each of the two beams while SHG from each individual beam is not phase-matched. This background-free set-up allows for large dynamic ranges, but is typically not the preferred set-up in the sub-10 fs regime. In a noncollinear set-up, an additional problem occurs due to beam smearing caused by the finite crossing angle of the two beams [95]. Therefore a collinear set-up is preferred. This then yields the interferometric autocorrelation trace [96]: IAC(δt) = [E(t) + E(t δt)] 2 2 dt. (6) Unfortunately, no way exists to retrieve the original pulse profile from any measured autocorrelation traces without additional knowledge. Inspired by a theoretical description of the mode-locking process, one can sometimes assume a certain pulse shape and then retrieval is simple. This is not an option in the sub-10 fs regime with its complex pulse shapes. In this regime, simple analytical functions must not be assumed anymore for reconstructing the original pulse shape from the measured autocorrelation function. Additionally, the sub- 10 fs regime is very demanding, and pulse shaping by spectral filtering or dispersion in the beam splitters and nonlinear crystal have to be kept to a minimum. Wherever possible, this regime calls for the use of metal-coated reflective optics. Methods have been discussed to solve the problem of retrieving the pulse shape that gave rise to a particular autocorrelation [24, 98]. We refer to these methods as decorrelation. Decorrelation methods require additional experimental information. In the simplest form, this information can be provided by asimultaneous measurement of the power spectrum of the laser, even though this has been shown not to fully remove ambiguities. Decorrelation methods employ a computer optimization strategy to find a simultaneous fit to measured spectrum and autocorrelation. This removes the arbitrariness of assuming a particular pulse shape for pulse retrieval, but requires data withexcellent signal-to-noise ratio for reliable operation. A practical example based on decorrelation of the IAC trace (equation (6)) and the spectrum is shown infigure Frequency-resolved optical gating and sonogram Even though invented earlier, one can conceptually understand frequency-resolved optical gating (FROG, [98, 99]) and the R7

8 Figure 6. Pulse characterization methods. From top to bottom: autocorrelation. The input pulse is split into two identical replicas. One of them serves as the reference sample and is temporarily delayed relative to the other. Multiplication of both replicas in a process such as second-harmonic generation or two-photon absorption delivers the autocorrelation function versus delay time shown on the right. This allows a coarse estimation of the pulse duration. FROG additionally spectrally disperses the autocorrelation function and delivers a wavelength-resolved autocorrelation function called the FROG trace (equation (7)). Other than simple autocorrelation, FROG allows complete reconstruction of the input pulse profile. Related is the sonogram technique, which spectrally resolves one of the input replicas instead (equation (9)). SPIDER also creates two replicas of the input pulse at a fixed delay and mixes these two with a chirped copy of the input pulse. The resulting two up-converted replicas are spectrally sheared with respect to each other. The resulting spectral interference pattern S(λ) allows reconstruction of the spectral phase of the input pulse. sonogram technique [100, 101] as a further extension of the decorrelation methods. For FROG, the autocorrelation of equation (5) is spectrally resolved for each and every delay step (see figure 6). The autocorrelation is then sampled on a {δτ,ω} grid. The autocorrelation spectrogram of the electric field E of the pulse + IFROG SHG (δt,ω)= 2 E(t)E(t δt) exp( iωt) dt (7) is called the SHG-FROG trace. Beyond SHG-FROG, which can be simply understood as an extension of autocorrelation methods, a wide variety of FROG methods have been described. Most notable is self-diffraction FROG, which is very important for measurements on amplifier systems [102]. Here the FROG trace is given by + IFROG SD (δt,ω)= 2 E(t) 2 E(t δt) exp( iωt) dt. (8) Its main advantage is that it cannot suffer from limited phasematching bandwidths as the SHG variant does. Self-diffraction FROG, however, requires significantly higher pulse energies and cannot be used for the oscillator-level pulse energies. The sonogram technique is very similar to FROG, but cuts out a spectral slice of one of the two replica pulses and cross-correlates it with the other, yielding the sonogram trace [100, 101] + 2 I Sonogram (δt,ω)= E( )F( ω) exp( i t) d. (9) R8

9 Figure 7. Examples of measurements of pulses from a Ti:sapphire laser. Top: SPIDER measurement and reconstructed pulse [116]. Middle: FROG trace and reconstructed pulse measured under nearly identical conditions at the same laser [108]. Bottom: iteratively reconstructed pulse shape from interferometric autocorrelation and power spectrum of the laser [20]. Note that all three measurements gave compatible pulse durations of about 6 fs or slightly below. Here F( ω) is the electric field of the filtered slice which is correlated with an unfiltered replica of the pulse under investigation. Both techniques, FROG and sonogram, record data on a two-dimensional array, rather than recording two onedimensional data traces as in the decorrelation. Examples for measurements are shown in figure 7. The excess data give rise to an increased robustness of the two-dimensional methods. Consequently, both methods show an improved immunity towards measurement noise. Moreover, FROG provides built-in consistency checks (marginals), which allow one to detect experimental flaws, e.g. due to limited phase-matching bandwidth or spectral filtering in the set-up. FROG can also be used in a cross-correlation variant to characterize one unknown pulse with the aid of another known one. This method is called XFROG and is of particular interest for the characterization of pulses in the UV and IR spectral range [103, 104]. This variant of FROG hasbeen demonstrated with the extremely complex pulse shapes of white-light continua generated in microstructure fibres [105]. Similar to the one-dimensional methods, FROG and the sonogram technique use an optimization strategy for pulse retrieval. Mathematically, it can be shown that knowledge R9

10 Figure 8. Ultrafast amplitude and phase modulators. Top: liquid crystal phase and amplitude shaper. The input pulse is spectrally dispersed and imaged onto the array with an adjustable retardation/absorption profile. This allows, for example, the generation of controllable pulse sequences from an input pulse. The same function can also be achieved by an acousto-optic device (middle). An ultrasonic wave is written into an acousto-optic deflector, where control of the acoustic waveform allows control of the deflection of the optical Fourier components into the first order of the device. Phase control can also be achieved with micro-machined mirror membranes, which can be electrostatically controlled (bottom). of the FROG trace of a pulse defines the pulse s amplitude and phase. There are some known cases of ambiguities, e.g. a time reversal ambiguity in the case of SHG FROG, which can be easily removed with an additional measurement. Apart from these special cases, which require some attention or special treatment, the reconstruction is effectively unambiguous. In contrast to the one-dimensional case, a well-defined solution for the decorrelation always exists, even though it can be timeconsuming for the algorithm to find this solution. Recent improvements of the FROG technique have lead to very sophisticated retrieval procedures, which can rapidly retrieve the pulse from the FROG trace and allow update rates up to several hertz [106]. In the sub-10 fs range, FROG has been demonstrated down to 4.5 fs pulse duration [107]. Use of FROG in this regime requires extremely careful design. Other than autocorrelation techniques, FROG offers the marginals as built-in consistency checks to detect problems caused by limited conversion bandwidth or spectral filtering. This is of particular importance in the sub-10 fs range. In this regime, the non-collinear geometry used inthe FROG set-up can also introduce beam-smearing artifacts. It is a particular challenge to design a FROG apparatus to keep these effects at a negligible level. Aperturing has been proposed as one method to eliminate these problems; use of a collinear set-up using type-ii phase matching can be an alternative and does not reduce signal strength [108] Spectral phase interferometry for direct electric field reconstruction All techniques described so far involve auto- or crosscorrelation together with spectral resolution to remove the ambiguity in pulse reconstruction. Recently, a completely different technique based on spectral interferometry [109] emerged for the characterization of femtosecond pulses. This technique is called spectral phase interferometry for direct electric-field reconstruction (SPIDER, [110]). The spectrum S(ω) of two identical pulses I (t) with respective temporal delay T is the spectrum of thesingle pulse Ĩ (ω) multiplied with a spectrally oscillating term. Measuring the spectral fringe spacing ω = 2π/ T of S(ω) allows us to determine the temporal spacing T of the two pulses. If these two pulses are identical, the fringe spacing of S(ω) is also strictly constant over the entire spectrum. A spectral interferogram between a chirped and an unchirped pulse, however, allows us not only to determine the delay between the pulses but also the difference in chirp between the two pulses. This R10

11 is the fundamental idea of spectral interferometry. SPIDER generates two delayed replicas of the pulse to be measured (see figure 6). It also generates a third pulse from the input pulse. The third is strongly chirped by sending it through a grating sequence or through a highly dispersive glass block. The dispersion creates a GD between the red and blue Fourier components of the third pulse (compare equations (2), (3)). This chirped pulse is then used to frequency-shift the two replicas of the input pulse using sum-frequency generation. Because of their temporal delay and the strong chirp on the up-converter pulse, both replicas are shifted in frequency by different amounts. Measuring the spectral interferogram of the up-converted replicas allows sampling of their relative phase delay as a function of frequency. This is now exactly the information needed to reconstruct the spectral phase. Together withanindependent measurement of the amplitude spectrum, this yields acomplete description of the pulse. Again, this technique is exemplified with pulses from a Ti:sapphire laser in figure 7. The SPIDER method has been demonstrated with sub- 6fspulses from Ti:sapphire lasers, compressed pulses from an amplifier system [111] and optical parametric amplifiers [37]. One of the major advantages of SPIDER is that it offers a direct reconstruction of the pulse profile, rather than requiring computationally intensive optimization strategies. Typically, acquisition speed is only limited by the readout speed of the CCD array used in the spectrograph. Acquisition and reconstruction rates of up to 20 Hz have been demonstrated [112], which makes SPIDER an ideal online tool for aligning complex femtosecond laser systems. SPIDER can also be used in combination with pulse shapers [113]. Other than evolutionary strategies, which try to compensate phase distortions of a pulse by optimizing, for example, its secondharmonic efficiency, SPIDER provides enough information to directly set the spectral phase to generate a bandwidthlimited pulse. Given the much moreconcise data of the SPIDERmethod, this should allow for a direct and much more rapid phase adaptation than lengthy evolutionary optimization strategies. The rapid data acquisition capabilities of the SPIDER method can also be exploited in another way to measure spatially resolved temporal pulse profiles. SPIDER can be adapted to spatially resolve measurements using an imaging spectrograph together with a two-dimensional array. This set-up spatially resolves temporal pulse profiles along the axis defined by the entrance slit of the spectrograph [114, 115]. Unlike the methods discussed so far which integrate over the spatial beam profile, spatial resolution enables the building of an ultrafast camera. Such an ultrafast camera can detect differences in pulse width between beam centre and off-centre regions. Several methods have been introduced that emulate the functionality of a simple photodiode well into the femtosecond range. The most concise diagnosis is offered by FROG and SPIDER, which set a new standard for the determination of pulse parameters. This additional information often provides valuable advice on possible adjustments of the dispersion compensation scheme employed [116, 117]. In [118], the phase of the output coupling mirror and the dispersion oscillations of the double-chirped mirrors can be identified to cause residual uncompensated phase aberrations on the output pulse. In [37] the roll-on of a transmission window of the chirped mirrors could be pinpointed as the cause for phase distortions. Baltuška and coworkers designed the mirrors, which were used inthegeneration of some of the shortest pulses generated to date, according to earlier measurements of the uncompressed pulse [107]. These examples clearly illustrate that knowledge beyond the pulse duration is useful to analyse limitations, which then enables us to push the pulse duration even further. The experimental constraints determine whether the FROG, sonogram or SPIDER method should be used. All these methods are very well adapted to the particular challenges of femtosecond pulse characterization. 5. Phase and amplitude modulation of short optical pulses Grating or prism sequences, such as those introduced in section 3, can also be employed to adjust dispersion in an adaptive way. A very common set-up is the so-called 4 f zero dispersion delay line [119, 120]. This set-up is very similar to the previously described stretcher, but operates at exactly 1magnification or an effective zero grating distance. A first grating disperses the input pulse. A lens at distance f from the grating then creates a spectrally dispersed picture of the input. A second identical lens grating system then reimages the spectrally dispersed picture back onto one point at distance f from the second grating. Provided there is proper alignment, this set-up is totally neutral in terms of dispersion, i.e. the shape of a pulse propagating through a 4 f assembly is not modified. However, as the pulse is spectrally dispersed in thefourier plane in the centre of the set-up, a phase or amplitude modulator array at this location allows manipulation of the waveform Liquid crystal arrays Several approaches exist for the technical implementation of the modulator array. These can be categorized into phase, amplitude and combined modulators. Another important aspect is pixelation and the number of pixels or degrees of freedom of such a device. Historically, liquid crystal arrays were first used in a 4 f shaper [119]. Liquid crystals can be used for phase modulators, and in combination with polarizers they may also serve as amplitude modulators. Combined devices consist of two liquid crystal arrays and polarizers and allow control of both amplitude and phase. Most devices are pixelated with pixel numbers between 128 and 640 [121, 122]. The pixels typically consist of a few 10 µm widestripes thatcanbeindividually electronically addressed. Unpixelated devices have also been reported, either addressed by an electron beam [123] or all-optically [124]. These devices may be preferred if satellite pulses stemming from the pixelation have to be avoided, even though these are typically not a concern. Liquid crystal phase masks have been used for a variety of applications. Used in a phase shaper, they can compensate for arbitrary dispersion. In this regard, it can be understood as a programmable microstructured dispersive device. It is particularly useful when dispersion is not very well known or may change over time. Recently, for example, an adaptive pulse compressor was used to compress pulses to sub-6 fs pulse duration [125, 126]. Note that the phase R11